affine and polynomial processes

affine and polynomial processes - ETH E-Collection

Diss. ETH No. 19652
AFFINE AND POLYNOMIAL
PROCESSES
A dissertation submitted to
ETH ZURICH
for the degree of
Doctor Of Sciences
presented by
CHRISTA CUCHIERO
Dipl. Ing. TU Vienna
born 13 April 1983
citizen of Austria
accepted on the recommendation of
Prof. Dr. Josef Teichmann
examiner
Prof. Dr. Walter Schachermayer
co-examiner
Ass. Prof. Dr. Johannes Muhle-Karbe co-examiner
2011
Abstract
This thesis is devoted to the study of affine and polynomial processes. Both are
particular classes of continuous-time Markov processes, whose specific properties are determined by the following assumptions on the associated semigroups:
in the case of affine processes, the Fourier-Laplace transform of the marginal
distributions is supposed to depend in an exponential affine way on the initial
state; in the case of polynomial processes, the semigroup is assumed to map
polynomials into polynomials. This implies in particular that under moment
assumptions, affine processes are a subclass of polynomial processes.
Apart from some recent developments in multivariate stochastic volatility
modeling, affine processes have mainly been studied on the particular state
n
space Rm
+ × R , where a complete characterization was provided by Duffie,
Filipović, and Schachermayer [2003].
n
Building on the results of affine processes on Rm
+ × R , the aim of this
thesis is twofold: first, the analysis of affine processes on general state spaces,
in particular convex cones and the subclass of symmetric cones; second, the
study of analytically tractable extensions of the affine class, which leads to
the introduction of polynomial processes.
The first crucial property needed to study affine processes is the differentiability of the Fourier-Laplace transform with respect to time, which is
called regularity. We establish this property for all affine processes on any
state space. Beyond that, we prove that every affine process admits a càdlàg
version and is a semimartingale up to its lifetime.
In order to further characterize affine processes, we strengthen our assumptions on the state space and consider affine processes on convex cones
and in particular on symmetric cones. The latter setting contains the cone
of positive semidefinite matrices, which is – in view of multivariate stochastic volatility modeling – particularly relevant for practical applications. On
irreducible symmetric cones we then provide a full characterization of affine
processes, meaning that the parameters of the semimartingale characteristics
necessarily satisfy some admissibility conditions, which are also sufficient for
existence.
i
ii
Abstract
The second line of research, namely a generalization of affine processes,
is based on the observation that the expected value of a polynomial of an
affine process is again a polynomial in its initial value. Taking this property
as definition of a class of Markov processes, which we call polynomial, gives
rise to an analytically tractable extension of the affine class, for which the
calculation of moments only requires the computation of matrix exponentials.
The last part of the thesis is devoted to applications of affine and polynomial processes, where we study multivariate affine stochastic volatility models
and variance reduction techniques in Monte-Carlo simulations by exploiting
the properties of polynomial processes.
Kurzfassung
Das Thema dieser Dissertation sind affine und polynomiale Prozesse. Beide
sind bestimmte Klassen zeitkontinuierlicher Markov-Prozesse, deren spezifische Eigenschaften durch folgende Annahmen an die Halbgruppen bestimmt
sind: im Fall der affinen Prozesse hängt die Fourier-Laplace-Transformierte
der Randverteilungen exponentiell affin vom Anfangszustand ab, während im
Fall der polynomialen Prozesse die Halbgruppe Polynome auf Polynome abbildet, d.h., der Erwartungswert eines jeden Polynoms ist wieder ein Polynom
im Anfangwert. Insbesondere bedeutet dies, dass affine Prozesse unter Momentenbedingungen eine Unterklasse der polynomialen Prozesse bilden.
Abgesehen von neueren Entwicklungen im Bereich multivariater Volatilitätsmodellierung wurden affine Prozesse hauptsächlich auf dem Zustandsraum
n
Rm
+ ×R untersucht und auf diesem von Duffie et al. [2003] vollständig charakterisiert.
Aufbauend auf diesen Ergebnissen ist das Ziel dieser Arbeit, einerseits
die Erforschung affiner Prozesse auf allgemeinen Zustandsräumen, insbesondere auf konvexen und symmetrischen Kegeln, und andererseits die Untersuchung analytisch gut handhabbarer Erweiterungen der affinen Klasse, was
zur Einführung polynomialer Prozesse führt.
Die erste entscheidende Eigenschaft für die Analysis affiner Prozesse ist
die Differenzierbarkeit der Fourier-Laplace-Transformierten bezüglich der Zeit,
was als Regularität bezeichnet wird. Wir zeigen diese Eigenschaft für alle
affinen Prozesse auf jedem beliebigen Zustandsraum. Darüber hinaus beweisen
wir, dass jeder affine Prozess eine càdlàg Version besitzt und ein Semimartingal
bis zum Ende seiner Lebenszeit ist.
Zur weiteren Charakterisierung affiner Prozesse treffen wir stärkere Annahmen an den Zustandsraum und schränken uns auf konvexe und symmetrische Kegel ein. Letztere enthalten den Kegel der positiv semidefiniten
Matrizen, der - in Anbetracht multivariater stochastischer Volatilitätsmodelle
- von besonderer Bedeutung für die Praxis ist. Auf irreduziblen symmetrischen
Kegeln erhalten wir eine vollständige Charakterisierung affiner Prozesse, d.h.,
die Parameter der Semimartingalcharakteristiken erfüllen notwendigerweise
iii
iv
Kurzfassung
bestimmte Zulässigkeitsbedingungen, die sich auch als hinreichend in Hinblick
auf die Existenz affiner Prozesse erweisen.
Die zweite Forschungsrichtung, nämlich eine Verallgemeinerung affiner Prozesse, beruht auf der Beobachtung, dass der Erwartungswert eines Polynoms
eines affinen Prozesses wieder ein Polynom in dessen Anfangswert ist. Definiert
man nun polynomiale Prozesse über diese Eigenschaft, führt dies zu einer
neuen Prozessklasse, für die Momente auf besonders einfache Weise durch
Matrixexponentiale berechnet werden können.
Der letzte Teil der Arbeit widmet sich Anwendungen von affinen und
polynomialen Prozessen. In diesem Zusammenhang analysieren wir multivariate affine stochastische Volatilitätsmodelle und Techniken zur Varianzreduktion in Monte-Carlo-Simulationen, wobei wir die Eigenschaften polynomialer
Prozesse ausnutzen.
Acknowledgment
First of all, I would like to express my gratitude to my advisor, Josef Teichmann, whose expertise, enthusiasm for research, understanding and encouragement added considerably to the completion of my thesis. I appreciate
not only his mathematical comprehension and vast knowledge in many areas, but also his kindness and amicable way of mentoring. Without doubt
he aroused my joy of mathematical research, inspired me with his invaluable
ideas, motivated me with fascinating discussions and enabled me to deepen
my understanding of stochastic analysis and mathematical finance.
My special thanks also go to Damir Filipović, my co-supervisor, who provided me with helpful advice and very in-depth inputs while working in Vienna. The collaboration with him considerably enriched my graduate experience and led to a very fruitful scientific work.
I would also like to thank Walter Schachermayer, who was head of the
Institute for Mathematical Methods in Economics and the Research Group for
Financial and Actuarial Mathematics (FAM) at TU Vienna, for the excellent
working environment and pleasant atmosphere at FAM. Beyond that, I am
also very grateful to him for refereeing my thesis. Likewise, I also want to
thank Johannes Muhle-Karbe for accepting the task of writing a report for
my thesis and providing me with very helpful suggestions.
Furthermore, I would particularly like to express my thanks to my coauthors Martin Keller-Ressel and Eberhard Mayerhofer. Our common work
and fruitful discussions constituted a real source of motivation throughout the
years of my graduate studies. Likewise, I also want to thank Elisa Nicolato
and David Skovmand, who opened new doors and perspectives to some applications of our field of research. Our time spent together, at and outside work,
has always been a great pleasure. I sincerely hope that these collaborations
will continue in the future.
Moreover, I am especially indebted to all my friends and colleagues at the
research groups at TU Vienna and ETH Zürich, which were both from a scientific and personal point of view perfect environments for my research. Seeing
truly good friends every day made the hard periods during my PhD studies a
v
vi
Acknowledgment
lot easier. My thanks go to my roommates in Vienna, Sara Karlsson, Antonis Papapantoleon, Takahiro Tsuchiya and to my colleagues in Zürich, Ozan
Akdogan, David Belius, Matteo Casserini, Christoph Czichowsky, Philipp
Dörsek, Nicoletta Gabrielli, Selim Gokay, Georg Grafendorfer, Martin Herdegen, Blanka Horvath, Florian Leisch, Marcel Nutz, Anja Richter, Stefan Tappe
and Dejan Veluscek. Thank you all for the coffees, beers, laughs, mathematical discussions and long evening sessions.
Finally, I gratefully acknowledge the financial support by ETH Zürich and
the START price project at TU Vienna.
In concluding, I would like to thank my friends and my family, in particular
Renaud, who has always patiently listened to my problems of mathematical
and non-mathematical nature and whose loving company, support and encouragement over the last years considerably helped me to accomplish the task of
writing this thesis.
Contents
Abstract
i
Kurzfassung
iii
Acknowledgment
v
Introduction
1
I
7
Affine Processes
1 Affine Processes on General State Spaces
1.1 Definition . . . . . . . . . . . . . . . . . .
1.2 Càdlàg Version . . . . . . . . . . . . . . .
1.3 Right-Continuity of the Filtration . . . . .
1.4 Semimartingale Property . . . . . . . . . .
1.5 Regularity . . . . . . . . . . . . . . . . . .
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2 Affine Processes on Proper Convex Cones
2.1 Definition of Cone-valued Affine Processes .
2.2 Feller Property and Regularity . . . . . . . .
2.3 Necessary Conditions . . . . . . . . . . . . .
2.3.1 Lévy Khintchine Form of F and R .
2.3.2 Parameter Restrictions . . . . . . . .
2.3.3 Quasi-monotonicity . . . . . . . . . .
2.4 The Generalized Riccati Equations . . . . .
2.5 Construction of Affine Pure Jump Processes
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3 Affine Processes on Symmetric Cones
3.1 Symmetric Cones and Euclidean Jordan Algebras . . . . . . .
3.2 Refinement of the Necessary Conditions . . . . . . . . . . . . .
3.2.1 Representation of the Diffusion Part . . . . . . . . . .
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vii
viii
Contents
3.3
3.4
3.5
3.6
3.7
II
3.2.2 Linear Jump Coefficient . . . . . . . . . . . . . .
3.2.3 The Special Role of the Constant Drift Part . . .
Discussion of the Admissibility Conditions . . . . . . . .
3.3.1 Diffusion . . . . . . . . . . . . . . . . . . . . . . .
3.3.2 Drift . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.3 Killing . . . . . . . . . . . . . . . . . . . . . . . .
3.3.4 Jumps . . . . . . . . . . . . . . . . . . . . . . . .
Construction of Affine Processes on Symmetric Cones . .
3.4.1 Construction of Affine Diffusion Processes . . . .
3.4.2 Existence of Affine processes on Symmetric cones
Wishart Distribution . . . . . . . . . . . . . . . . . . . .
3.5.1 Central Wishart Distribution . . . . . . . . . . .
3.5.2 Non-central Wishart distribution . . . . . . . . .
Relation to Infinitely Divisible Distributions . . . . . . .
Results for Positive Semidefinite Matrices . . . . . . . . .
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Polynomial Processes
4 Characterization and Relation to Semimartingales
4.1 Definition and Characterization . . . . . . . . . . . . . .
4.2 Polynomial Semimartingales . . . . . . . . . . . . . . . .
4.3 Characterization by means of the Extended Generator .
4.3.1 Proof of Theorem 4.1.8 (iv) ⇒ (iii) . . . . . . . .
4.3.2 Semimartingales which are Polynomial Processes
4.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . .
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Applications
5 Multivariate Affine Volatility Models
5.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Form of F and R . . . . . . . . . . . . . . . . . . . . . . .
5.3 Conservativeness and Martingale Property . . . . . . . . .
5.4 Semimartingale Representation of Affine Volatility Models
5.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5.1 Multivariate Heston Model . . . . . . . . . . . . . .
5.5.2 Multivariate Barndorff-Nielsen-Shepard Model . . .
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Contents
ix
6 Applications of Polynomial Processes
6.1 Moment Calculation . . . . . . . . . . . . . . . . . . . . . . .
6.1.1 Generalized Method of Moments . . . . . . . . . . . .
6.2 Pricing - Variance Reduction . . . . . . . . . . . . . . . . . . .
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Appendix
173
A Symmetric Cones and Euclidean Jordan Algebras
A.1 Important Definitions . . . . . . . . . . . . . . . . . . . . .
A.1.1 Determinant, Trace and Inverse . . . . . . . . . . .
A.1.2 Idempotents, Spectral and Peirce Decomposition . .
A.1.3 Classification of Simple Euclidean Jordan Algebras
A.2 Some Results . . . . . . . . . . . . . . . . . . . . . . . . .
Bibliography
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Introduction
The aim of this thesis is to draw a comprehensive picture of the theory of
affine and polynomial processes. Both are particular classes of continuous-time
Markov processes, whose specific properties are determined by the following
assumptions on the associated semigroups: in the case of affine processes, the
Fourier-Laplace transform of the marginal distributions is supposed to depend
in an exponential affine way on the initial state; in the case of polynomial
processes, the semigroup is assumed to map polynomials into polynomials,
that is, the expected value of any polynomial is again a polynomial in the
initial state. This implies in particular that under moment assumptions, affine
processes are a subclass of polynomial processes.
The common property of both kinds of processes is the high degree of analytical tractability implied by the fact that the expected value of a large class
of payoff functions is explicitly known. This is one reason why many models
used in mathematical finance fall under the setting of affine or polynomial
processes. Moreover, as a particular class of jump diffusions, they allow for simultaneous modeling of diffusive, jump and stochastic volatility phenomena in
a multivariate setting, while remaining analytically tractable in a remarkable
way.
From a mathematical point of view the affine class contains Lévy processes,
Ornstein-Uhlenbeck processes, continuous branching processes and their multidimensional generalizations such as matrix-valued Wishart processes. Beyond that, processes with quadratic diffusion coefficients, such as the Jacobi process or Lévy-driven stochastic differential equations with affine vector
fields, are covered by the setting of polynomial processes.
The analysis of affine processes dates back to 1971, when Kawazu and
Watanabe [1971] studied continuous-time limits of Galton-Watson branching
processes with immigration. These processes correspond exactly to the onedimensional affine processes on the positive real line, which were taken up later
in the short-rate model of Cox, Ingersoll, and Ross [1985] and the stochastic
volatility model of Heston [1993]. The need for more complex models pro1
2
Introduction
gressively led to the introduction of higher dimensional affine jump diffusions
n−m
, see, e.g., Dai and
on the so-called canonical state space, that is, Rm
+ ×R
Singleton [2000], Duffie and Kan [1996], Duffie, Pan, and Singleton [2000].
These affine jump diffusions are included in the setting of affine processes
n−m
on Rm
, which are defined as continuous-time Markov processes with
+ ×R
the mentioned exponential affine property of the characteristic function. A
complete characterization of these processes in terms of semimartingale characteristics and Markov generators was subsequently provided by Duffie et al.
[2003].
The development of multivariate stochastic volatility models has recently
given rise to applications of affine processes on non-canonical state spaces, in
particular on the cone of positive semidefinite matrices. Such matrix-valued
affine processes seem to have been studied systematically for the first time
in the literature by Bru [1989, 1991], who introduced the so-called Wishart
processes, which are multidimensional analogs of squared Bessel processes.
A natural generalization of positive semidefinite matrices are so-called symmetric cones, which are cones of squares in Jordan algebras (see the work
of Faraut and Korányi [1994]). Affine diffusion processes thereon were introduced by Grasselli and Tebaldi [2008] with the objective to give some general
results on the solvability of the corresponding Riccati differential equations.
The setting of symmetric cones is taken up in this thesis.
State spaces whose boundary is described by a quadratic polynomial have
been considered by Spreij and Veerman [2010]. It turns out that up to isomorphisms the only possible state spaces of this form are parabolic ones (see
also Duffie et al. [2003, Section 12]) and the symmetric Lorentz cone. Let us
remark that the boundary of other symmetric cones is in general described by
polynomials of higher degree.
Apart from these recent developments, affine processes have mainly been
n−m
studied on the particular state space Rm
. Building on the results
+ × R
of affine processes on the canonical state space, the purpose of this thesis is
twofold: first, the analysis of affine processes on general state spaces, in particular convex cones and the subclass of symmetric cones; second, the study
of analytically tractable extensions of the affine class, which leads to the introduction of polynomial processes.
As mentioned before, the first line of research is motivated by multivariate
affine stochastic volatility models, which consist of a d-dimensional logarithmic
price process Y and an instantaneous stochastic covariation process X. These
models correspond to particular affine processes on the mixed state space
Sd+ × Rd , where Sd+ denotes the cone of positive semidefinite d × d matrices.
The first crucial property needed to study such models is the differentiabil-
3
ity of the Fourier-Laplace transform of the joint process (X, Y ) with respect to
time, which is called regularity. This implies in particular that the (extended)
cumulant generating function of (X, Y ) is given as solution of a system of ordinary differential equations, upon which the analytical tractability of affine
stochastic volatility models is based. The main theorem of Chapter 1 establishes regularity for all affine processes on any state space. An alternative proof
of this result was obtained by Keller-Ressel, Schachermayer, and Teichmann
[2011]. The same authors also proved regularity on the canonical state space
(see Keller-Ressel, Schachermayer, and Teichmann [2010]) by using different
n−m
methods which exploit the structure of Rm
and cannot be extended
+ ×R
to general state spaces.
On the way to this result, we show additionally that every affine process
admits a càdlàg version and is a semimartingale up to its lifetime, whose
(absolutely continuous) characteristics depend in an affine way on the state
variables. This implies in particular that the derivative of the Fourier-Laplace
transform at t = 0 is of Lévy-Khintchine form. The results of this chapter
will be published in Cuchiero and Teichmann [2011].
In order to further characterize affine processes, in particular in view of the
question of existence, we assume the state space to be a closed convex cone in
Chapter 2. This allows us to prove the Feller property and existence of affine
pure jump processes without diffusion component. The latter result relies on
the global existence and uniqueness of the corresponding generalized Riccati
differential equations, by means of which the Fourier-Laplace transform is
determined. Indeed, in the case of affine pure jump processes the solutions of
these equations can be recognized as cumulant generating functions of substochastic measures, as done in Duffie et al. [2003, Section 7]. This then settles
the existence question.
For the characterization of general affine processes with diffusion component we further strengthen the assumptions on the state space by assuming
the setting of symmetric cones. This framework contains the cone of positive
semidefinite matrices, which is – in view of multivariate stochastic volatility
modeling – particularly relevant for practical applications. Other examples
are Rm
+ , the Hermitian matrices or the Lorentz cone. This assumption on the
state space allows us to provide a full characterization of affine processes on irreducible symmetric cones. Our results of Chapter 3 show that the parameters
of the semimartingale characteristics satisfy some well-determined admissibility conditions, which differ in particular with regard to the constant drift part
from those on the canonical state space. Conversely, and more importantly
for applications, we show that for any admissible parameter set there exists a
unique well-behaved affine process.
In the case of positive semidefinite matrices, where affine processes have
4
Introduction
already been studied in the context of stochastic volatility models, our findings
extend the model class since a more general drift and jumps are possible. On
the other hand, we also establish the exact assumptions under which affine
processes on Sd+ actually exist.
The existence proof is based on the result that the solutions of the generalized Riccati differential equations of a pure diffusion process with a particular
drift can be recognized as cumulant generating functions of the non-central
Wishart distribution. This distribution is well studied on the cone of positive
semidefinite matrices (see, e.g., Letac and Massam [2004]), but for general
symmetric cones only a few results on the central Wishart distribution are
available in the literature. For example, the form of the density function
has not been provided so far and is derived in Section 3.5. Beyond that,
we prove that any stochastically continuous infinitely decomposable Markov
process on a symmetric cone, whose dimension is greater than 2, is affine
with zero diffusion, and vice versa. This is a consequence of the fact that the
Wishart distribution is not infinitely divisible. The findings of Chapter 2 and 3
generalize the results on positive semidefinite matrices obtained in Cuchiero,
Filipović, Mayerhofer, and Teichmann [2011a] to the setting of (symmetric)
cones and will be published in Cuchiero, Keller-Ressel, Mayerhofer, and Teichmann [2011b]. For reasons of practical relevance and potential applications
in mathematical finance, we restate the main theorems in the specific context
of positive semidefinite matrices (see Section 3.7). This also summarizes the
results found in Cuchiero et al. [2011a], of which some parts are now proved
differently. In particular, in Cuchiero et al. [2011a] the question of existence
of affine processes is handled by solving the associated martingale problem
(see Cuchiero et al. [2011a, Proposition 5.9]), whereas it is here reduced via
Trotter’s product formula to the existence of pure diffusion and pure jump
processes.
The second line of research, namely a generalization of affine processes,
is based on the observation that the expected value of a polynomial of an
affine process is again a polynomial in the initial value. Taking this property
as definition of a class of Markov processes, which we call polynomial, gives
rise to an analytically tractable extension of the affine class, containing for
example processes with quadratic diffusion coefficients. We can thus describe
a class of processes for which it is easy and efficient to compute moments of all
orders, even though neither their probability distribution nor their characteristic function needs to be known. Indeed, the calculation of (mixed) moments
only requires the computation of matrix exponentials, where the entries of the
corresponding matrix can be easily deduced from the (extended) infinitesimal
generator. This property of polynomial processes has already been exploited
5
by Zhou [2003] for GMM estimation of one-dimensional jump-diffusions. In
Chapter 4 we formally introduce this class of polynomial processes, establish
a relationship to semimartingales and give conditions on the (extended) infinitesimal generator such that a Markov process is polynomial. This chapter
will be published in Cuchiero, Keller-Ressel, and Teichmann [2010].
The last part of the thesis is devoted to applications of affine and polynomial processes. In Chapter 5 we study multivariate affine stochastic volatility
models, where we establish necessary and sufficient conditions on the parameters describing the semimartingale characteristics such that the (discounted)
price processes are martingales. In particular, we prove the necessity of a
certain correlation structure between the Brownian motions driving the logarithmic price processes and the covariation process. The results of this chapter
will be published in Cuchiero [2011]. Chapter 6 then deals with applications
of polynomial processes, especially how the analytical knowledge of moments
can be exploited for option pricing. This chapter is part of Cuchiero et al.
[2010].
Notation
For the stochastic background and notation we refer to standard text books
such as Jacod and Shiryaev [2003] and Revuz and Yor [1999].
We write R+ for [0, ∞), R++ for (0, ∞) and Q+ for nonnegative rational
numbers. Moreover, V always denotes some finite dimensional real vector
space with scalar product h·, ·i. The symmetric matrices and the positive
semidefinite matrices over V are denoted by S(V ) and S+ (V ), respectively,
while L(V ) corresponds to the space of linear maps on V . In the case V = Rd ,
we write Sd and Sd+ for S(V ) and S+ (V ), respectively, and Id denotes the d×d
identity matrix.
Throughout this thesis, we shall consider the following function spaces for
measurable U ⊆ V , whose Borel σ-algebra is denoted by B(U ). We write
C(U ) for the space of (complex-valued) continuous functions f on U , Cb (U )
for the space of (complex-valued) bounded continuous functions on U , Cc (U )
for the space of functions f ∈ C(U ) with compact support and C0 (U ) for
the Banach space of functions f ∈ C(U ) with limkxk→∞ f (x) = 0 and norm
kf k∞ = supx∈U |f (x)|.
Part I
Affine Processes
7
Chapter 1
Affine Processes on General
State Spaces
We define affine processes as a particular class of time-homogeneous Markov
processes with state space D ⊆ V , some closed, non-empty subset of an ndimensional real vector space V with scalar product h·, ·i. To clarify notation,
we find it useful to recall in the first section the basic ingredients of the
theory of time-homogeneous Markov processes and the particular conventions
being made in this thesis (compare Blumenthal and Getoor [1968, Chapter
1.3], Chung and Walsh [2005, Chapter 1.2], Ethier and Kurtz [1986, Chapter
4], Rogers and Williams [1994, Chapter 3, Definition 1.1]).
1.1
Definition
Throughout D denotes a closed subset of V and D its Borel σ-algebra. Since
we shall not assume the process to be conservative, we adjoin to the state
space D a point ∆ ∈
/ D, called cemetery state, and set D∆ = D ∪ {∆} as well
as D∆ = σ(D, {∆}). We make the convention that k∆k := ∞, where k · k
denotes the norm induced by the scalar product h·, ·i, and we set f (∆) = 0
for any other function f on D.
Consider the following objects on a space Ω:
(i) a stochastic process X = (Xt )t≥0 taking values in D∆ such that
if Xs (ω) = ∆, then Xt (ω) = ∆ for all t ≥ s and all ω ∈ Ω;
(1.1)
(ii) the filtration
generated by X, that is, Ft0 = σ(Xs , s ≤ t), where we set
W
F 0 = t∈R+ Ft0 ;
(iii) a family of probability measures (Px )x∈D∆ on (Ω, F 0 ).
9
10
Chapter 1. Affine Processes on General State Spaces
Definition 1.1.1 (Markov process). A time-homogeneous Markov process
X = Ω, (Ft0 )t≥0 , (Xt )t≥0 , (pt )t≥0 , (Px )x∈D∆
with state space (D, D) (augmented by ∆) is a D∆ -valued stochastic process
such that, for all s, t ≥ 0, x ∈ D∆ and all bounded D∆ -measurable functions
f : D∆ → R,
(1.2)
Ex f (Xt+s )|Fs0 = EXs [f (Xt )] , Px -a.s.
Here, Ex denotes the expectation with respect to Px and (pt )t≥0 is a transition
function on (D∆ , D∆ ). A transition function is a family of kernels pt : D∆ ×
D∆ → [0, 1] such that
(i) for all t ≥ 0 and x ∈ D∆ , pt (x, ·) is a measure on D∆ with pt (x, D) ≤ 1,
pt (x, {∆}) = 1 − pt (x, D) and pt (∆, {∆}) = 1;
(ii) for all x ∈ D∆ , p0 (x, ·) = δx (·), where δx (·) denotes the Dirac measure;
(iii) for all t ≥ 0 and Γ ∈ D∆ , x 7→ pt (x, Γ) is D∆ -measurable;
(iv) for all s, t ≥ 0, x ∈ D∆ and Γ ∈ D∆ , the Chapman-Kolmogorov equation
holds, that is,
Z
pt+s (x, Γ) =
ps (x, dξ)pt (ξ, Γ).
D∆
If (Ft )t≥0 is a filtration with Ft0 ⊂ Ft , t ≥ 0, then X is a time-homogeneous
Markov process relative to (Ft ) if (1.2) holds with Fs0 replaced by Fs .
We can alternatively think of the transition function as inducing a measurable contraction semigroup (Pt )t≥0 defined by
Z
Pt f (x) := Ex [f (Xt )] =
f (ξ)pt (x, dξ), x ∈ D∆ ,
D
for all bounded D∆ -measurable functions f : D∆ → R.
Remark 1.1.2. (i) Note that, in contrast to Duffie et al. [2003], we do not
assume Ω to be the canonical space of all functions ω : R+ → D∆ , but
work on some general probability space.
(ii) Since we have pt (x, Γ) = Px [Xt ∈ Γ] for all t ≥ 0, x ∈ D∆ and Γ ∈ D∆ ,
property (ii) and (iii) of the transition function, imply Px [X0 = x] = 1
for all x ∈ D∆ and measurability of the map x 7→ Px [Xt ∈ Γ] for all
t ≥ 0 and Γ ∈ D∆ .
1.1. Definition
11
For the following definition of affine processes, let us introduce the set U
defined by
U = u ∈ V + i V ehu,xi is a bounded function on D .
(1.3)
Clearly i V ⊆ U. Here, the set i V stands for purely imaginary elements and
h·, ·i is the extension of the real scalar product to V + i V without complex
conjugation. Moreover, we denote by p the dimension of Re U and write
hRe Ui for its (real) linear hull and hRe Ui⊥ for its orthogonal complement
in V . The set i hRe Ui⊥ ⊂ U are then the purely imaginary direction of U.
Finally, for some linear subspace W ⊂ V , ΠW : V → V denotes the orthogonal
projection on W , which is extended to V +i V by linearity, i.e., ΠW (v1 +i v2 ) :=
ΠW v1 + i ΠW v2 .
Assumption 1.1.3. Recall that dim V = n. We require that the state space
D contains at least n + 1 affinely independent elements x1 , . . . , xn+1 , that is,
the n vectors (x1 − xj , . . . , xj−1 − xj , xj+1 − xj , . . . , xn+1 − xj ) are linearly
independent for every j ∈ {1, . . . , n + 1}.
We are now prepared to give our main definition:
Definition 1.1.4 (Affine process). A time-homogeneous Markov process X
relative to some filtration (Ft ) and with state space (D, D) (augmented by ∆)
is called affine if
(i) it is stochastically continuous, that is, lims→t ps (x, ·) = pt (x, ·) weakly on
D for every t ≥ 0 and x ∈ D, and
(ii) its Fourier-Laplace transform has exponential-affine dependence on the
initial state. This means that there exist functions Φ : R+ × U → C and
Ψ : R+ × U → V + i V such that
Z
hu,Xt i hu,xi
Ex e
= Pt e
=
ehu,ξi pt (x, dξ) = Φ(t, u)ehΨ(t,u),xi ,
(1.4)
D
for all x ∈ D and (t, u) ∈ R+ × U.
Remark 1.1.5. (i) The above definition differs in three crucial details from
the definitions given in Duffie et al. [2003, Definition 2.1, Definition
12.1].1
First, therein the right hand side of (1.4) is defined in terms of a function
φ(t, u), namely as eφ(t,u)+hΨ(t,u),xi , such that the function Φ(t, u) in our
1
n−m
In Definition 2.1 affine processes on the canonical state space D = Rm
are
+ × R
considered, whereas in Definition 12.1 the state space D can be an arbitrary subset of Rn .
12
Chapter 1. Affine Processes on General State Spaces
definition corresponds to eφ(t,u) .2 Our definition is in line with the one
given in Kawazu and Watanabe [1971] and Keller-Ressel et al. [2010]
and differs from the one in Duffie et al. [2003], as we do not require
Φ(t, u) 6= 0 a priori. However, since all affine processes on D = Rm
+ ×
Rn−m are infinitely divisible (see Duffie et al. [2003, Theorem 2.15]),
it turns out with hindsight that setting Φ(t, u) = eφ(t,u) is actually no
restriction.
Second, we assume that the affine property holds for all u ∈ U, whereas
n−m
it is restricted to i Rn
on the canonical state space D = Rm
+ × R
(see Duffie et al. [2003]). This however turns out to imply the affine
property also on U.
Third, in contrast to Duffie et al. [2003], we take stochastic continuity as
part of the definition of an affine process. We remark that there are simple examples of Markov processes which satisfy Definition 1.1.4 (ii), but
are not stochastically continuous (see Duffie et al. [2003, Remark 2.11]).
(ii) Note that Assumption 1.1.3 is no restriction, since we can always pass
to a lower dimensional ambient vector space if D does not contain n + 1
affinely independent elements.
(iii) Let us remark that in Section 1.2 we consider affine processes on the
filteredWspace (Ω, F 0 , Ft0 ), where Ft0 denotes the natural filtration and
F 0 = t∈R+ Ft0 , as introduced above. However, we shall progressively
enlarge the filtration by augmenting with the respective null-sets.
Before deducing the first properties of Φ and Ψ from the above definition,
let us introduce the sets
m
hReu,xi
U = u ∈ V + i V | sup e
≤ m , m ≥ 1,
x∈D
and note that U =
S
m≥1
U m and i V ⊆ U m for all m ≥ 1.
Proposition 1.1.6. Let X be an affine process relative to some filtration (Ft ).
Then the functions Φ and Ψ have the following properties:
(i) For every m ≥ 1, Φ and Ψ can be chosen to be jointly continuous
on Qm = {(t, u) ∈ R+ × U m | Φ(s, u) 6= 0, for all s ∈ [0, t]}. This
then yields a unique specification of Φ and Ψ on Q = {(t, u) ∈ R+ ×
U | Φ(s, u) 6= 0, for all s ∈ [0, t]}.
2
Note that the function Ψ in our definition is denoted by ψ in Duffie et al. [2003,
Definition 2.1].
1.1. Definition
13
(ii) Ψ maps the set O = {(t, u) ∈ R+ × U | Φ(t, u) 6= 0} to U.
(iii) Φ(0, u) = 1 and Ψ(0, u) = u for all u ∈ U.
(iv) The functions Φ and Ψ satisfy the semiflow property: Let u ∈ U and
t, s ≥ 0. Suppose that Φ(t + s, u) 6= 0, then also Φ(t, u) 6= 0 and
Φ(s, Ψ(t, u)) 6= 0 and we have
Φ(t + s, u) = Φ(t, u)Φ(s, Ψ(t, u)),
Ψ(t + s, u) = Ψ(s, Ψ(t, u)).
(1.5)
Proof. It follows e.g. from Bauer [1996, Lemma 23.7] that stochastic continuity
of X implies joint continuity of (t, u) 7→ Pt ehu,xi on R+ × U m for all x ∈ D.
Hence (t, u) 7→ Φ(t, u)ehΨ(t,u),xi is jointly continuous on R+ × U m for every
x ∈ D. By Assumption 1.1.3 on the state space D, this in turn yields a
unique continuous choice of the functions (t, u) 7→ Φ(t, u) and (t, u) 7→ Ψ(t, u)
on Qm . (compare Keller-Ressel et al. [2011, Proposition 2.4] for details).
Concerning (ii), let (t, u) ∈ O = {(t, u) ∈ R+ × U | Φ(t, u) 6= 0}. Since
Φ(t, u)ehΨ(t,u),xi = Ex ehu,Xt i ≤ Ex ehu,Xt i is bounded on D and as Φ(t, u) 6= 0, we conclude that Ψ(t, u) ∈ U.
Assertion (iii) follows simply from
ehu,xi = Ex ehu,X0 i = Φ(0, u)ehΨ(0,u),xi .
Assumption Φ(t + s, u) 6= 0 in (iv) implies
Ex ehu,Xt+s i = Φ(t + s, u)ehΨ(t+s,u),xi 6= 0.
(1.6)
By the law of iterated expectations and the Markov property, we thus have
ii
h h
hu,Xt+s i hu,Xt+s i Ex e
= Ex Ex e
(1.7)
Fs = Ex EXs ehu,Xt i .
If Φ(t, u) = 0 or Φ(s, Ψ(t, u)) = 0, then the inner or the outer expectation
evaluates to 0. This implies that the whole expression is 0, which contradicts (1.6). Hence Φ(t, u) 6= 0 and Φ(s, Ψ(t, u)) 6= 0 and we can write (1.7)
as
Ex ehu,Xt+s i = Ex Φ(t, u)ehΨ(t,u),Xs i = Φ(t, u)Φ(s, Ψ(t, u))ehΨ(s,Ψ(t,u)),xi .
Comparing with (1.6) yields the claim.
Remark 1.1.7. Henceforth, the symbols Φ and Ψ always correspond to the
unique continuous choice established in Proposition 1.1.6.
14
Chapter 1. Affine Processes on General State Spaces
1.2
Càdlàg Version
The aim of this section is to show that the definition of an affine process already
implies the existence of a càdlàg version. Indeed, for every fixed x ∈ D, we
first establish that for Px -almost every ω
t 7→ MtT,u (ω) := Φ(T − t, u)ehΨ(T −t,u),Xt (ω)i ,
t ∈ [0, T ],
is the restriction to Q+ ∩ [0, T ] of a càdlàg function for almost all (T, u) ∈
(0, ∞)×U, in the sense that MtT,u = 0 if Φ(T −t, u) = 0. This is an application
of Doob’s regularity theorem for supermartingales, where we can conclude –
using Fubini’s theorem – that there exists a Px -null-set outside of which we
observe appropriately regular trajectories for almost all (T, u).
Proposition 1.2.1. Let x ∈ D be fixed and let X be an affine process relative
to (Ft0 ). Then
lim MqT,u = lim Φ(T − q, u)ehΨ(T −q,u),Xq i ,
q∈Q+
q↓t
q∈Q+
q↓t
t ∈ [0, T ],
exists Px -a.s. for almost all (T, u) ∈ (0, ∞) × U and defines a càdlàg function
in t.
Proof. In order to prove this result, we adapt parts of the proof of Protter
[2005, Theorem I.4.30] to our setting. Due to the law of iterated expectations
MtT,u = Φ(T − t, u)ehΨ(T −t,u),Xt i = Ex ehu,XT i Ft0 , t ∈ [0, T ],
is a (complex-valued) (Ft0 , Px )-martingale for every u ∈ U and every T >
0. From Doob’s regularity theorem (see, e.g., Rogers and Williams [1994,
Theorem II.65.1]) it then follows that, for any fixed (T, u), the function t 7→
MtT,u (ω), with t ∈ Q+ ∩ [0, T ], is the restriction to Q+ ∩ [0, T ] of a càdlàg
function for Px -almost every ω. Define now the set
Γ = {(ω, T, u) ∈ Ω × (0, ∞) × U | t 7→ MtT,u (ω), t ∈ Q+ ∩ [0, T ],
is not the restriction of a càdlàg function}. (1.8)
Then Γ is a F 0 ⊗ B((0, ∞) × U)-measurable set. Due
R to the above argument
concerning regular versions of (super-)martingales, Ω 1Γ (ω, T, u)Px (dω) = 0
for any (T, u) ∈ (0, ∞) × U. By Fubini’s theorem, we therefore have
Z Z
Z
Z
1Γ (ω, T, u)dλ Px (dω) =
1Γ (ω, T, u)Px (dω) dλ = 0,
Ω
(0,∞)×U
(0,∞)×U
Ω
where λ denotes the Lebesgue measure. Hence, for Px -almost every ω, t 7→
MtT,u (ω), t ∈ Q+ ∩ [0, T ], is the restriction of a càdlàg function for λ-almost
all (T, u) ∈ (0, ∞) × U, which proves the result.
1.2. Càdlàg Version
15
Having established path regularity of the martingales M T,u , we want to
deduce the same for the affine process X. This is the purpose of the subsequent
lemmas and propositions, for which we need to introduce the following sets
e ⊆ Ω, T ⊆ (0, ∞) and V ⊆ i V :
Ω
e is the projection of {Ω × (0, ∞) × i V } \ Γ onto Ω,
Ω
T is the projection of {Ω × (0, ∞) × i V } \ Γ onto (0, ∞),
V is the projection of {Ω × (0, ∞) × U} \ Γ onto U,
(1.9)
(1.10)
(1.11)
where Γ is given in (1.8). Denoting by F x the completion of F 0 with respect to
e ∈ Fx
Px , let us remark that the measurable projection theorem implies that Ω
e = 1. Finally, for some r > 0,
and by the above proposition we have Px [Ω]
we denote by K the intersection of V with the closed ball with center 0 and
radius r, that is,
(1.12)
K := B(0, r) ∩ V := u ∈ U | kReuk2 + kImuk2 ≤ r2 ∩ V.
Lemma 1.2.2. Consider the set K defined in (1.12) and the function Ψ given
in (1.4) with the properties of Proposition 1.1.6. Denote by p the dimension
of Re U. Let (u1 , . . . , up ) be linearly independent vectors in K ∩ Re U and let
(up+1 , . . . , un ) be linearly independent vectors in ΠhRe U i⊥ K. Then there exists
some δ > 0 such that for every t ∈ [0, δ)
(Ψ(t, u1 ), . . . , Ψ(t, up ))
and
(ΠhRe U i⊥ Ψ(t, up+1 ), . . . , ΠhRe U i⊥ Ψ(t, un ))
are linearly independent.
Proof. This is simply a consequence of the fact that Ψ(0, u) = u for all u ∈
U ⊃ K and the continuity of t 7→ Ψ(t, u).
The following lemma is needed to prove Proposition 1.2.4 below which is
essential for establishing the existence of a càdlàg version of X.
Lemma 1.2.3. Let Ψ be given by (1.4) and assume that there exists some
D-valued sequence (xk )k∈N such that
lim ΠhRe U i xk =: lim yk
(1.13)
lim sup kΠhRe U i⊥ xk k = ∞.
(1.14)
k→∞
k→∞
exists finitely valued and
k→∞
16
Chapter 1. Affine Processes on General State Spaces
Then we can choose a subsequence of (xk ) denoted again by (xk ): along this
sequence there exist a finite number of mutually orthogonal directions gi ∈
hRe Ui⊥ of length 1 such that
X
xk −
hxk , gi igi
i
converges as k → ∞ and hxk , gi i diverges as k → ∞, where the rates of divergence are decreasing in i (see the proof for the precise statement). Furthermore, there exist continuous functions R : R+ → R++ and λi : R+ → hRe Ui⊥
such that
hΨ(t, u), gi i = hλi (t), ui
for all u ∈ ΠhRe U i⊥ U with kImuk < R(t).
Proof. Concerning the first assertion, we define – by choosing appropriate
subsequences, still denoted by (xk ), – the directions of divergence in hRe Ui⊥
inductively by
P
xk − r−1
i=1 hxk , gi igi
,
(1.15)
gr = lim
Pr−1
k→∞ kxk −
hx
,
g
ig
k
k
i
i
i=1
Pr−1
as long as lim supk→∞ kxk − i=1 hxk , gi igi k = ∞. Notice that we can choose
the directions gi mutually orthogonal and the rates of divergence of hgi , xk i
decreasing in i.
For the second part of the statement, we adapt the proof of Keller-Ressel
et al. [2010, Lemma 3.1] to our situation, using in particular the existence
of a sequence in D with the properties (1.13) and (1.14). As characteristic
function, the map i V 3 u 7→ Ex [ehu,Xt i ] is positive definite for any x ∈ D and
t ≥ 0. Define now for every u ∈ ΠhRe U i⊥ U ⊆ i V , x ∈ D and t ≥ 0 the function
Ex ehu,Xt i
Φ(t, u)ehΨ(t,u),xi
=
. (1.16)
Θ(u, t, x) =
Φ(t, 0)ehΠhRe U i Ψ(t,0),ΠhRe U i xi
Φ(t, 0)ehΠhRe U i Ψ(t,0),ΠhRe U i xi
As Ex eh0,Xt i = Φ(t, 0)ehΨ(t,0),xi is real-valued and positive for all t ≥ 0,
we conclude – due to Assumption 1.1.3 and the continuity of the functions
t 7→ Φ(t, 0) and t 7→ Ψ(t, 0) – that ImΦ(t, 0) = 0 and ImΨ(t, 0) = 0 for all
t ≥ 0. In particular, the denominator in (1.16) is positive, which implies that
i V ⊇ ΠhRe U i⊥ U 3 u 7→ Θ(u, t, x) is a positive definite function for all t ≥ 0
and x ∈ D. Moreover, since ΠhRe U i⊥ Ψ(t, 0) is purely imaginary and thus in
particular 0 for all t ≥ 0, it follows that
Θ(0, t, x) = exp hΠhRe U i⊥ Ψ(t, 0), ΠhRe U i⊥ xi = 1
1.2. Càdlàg Version
17
for all t ≥ 0 and x ∈ D. This together with the positive definiteness of
u 7→ Θ(u, t, x) yields
|Θ(u + v, t, x) − Θ(u, t, x)Θ(v, t, x)|2 ≤ 1,
u, v ∈ ΠhRe U i⊥ U, t ≥ 0, x ∈ D,
(1.17)
(compare, e.g., Keller-Ressel et al. [2010, Lemma 3.2]). Let us now define
y := ΠhRe U i x and
Φ(t, u + v)ehΠhRe U i Ψ(t,u+v),yi
,
Φ(t, 0)ehΠhRe U i Ψ(t,0),yi
Φ(t, u)Φ(t, v)ehΠhRe U i (Ψ(t,u)+Ψ(t,v)),yi
Z2 (u, v, y, t) =
,
Φ(t, 0)2 e2hΠhRe U i Ψ(t,0),yi
β1 (u, v, t) = Im(ΠhRe U i⊥ Ψ(t, u + v)),
Z1 (u, v, y, t) =
β2 (u, v, t) = Im(ΠhRe U i⊥ Ψ(t, u)) + Im(ΠhRe U i⊥ Ψ(t, v)),
Φ(t, u + v) hRe(Π
hRe U i (Ψ(t,u+v)−Ψ(t,0))),yi
e
r1 (u, v, y, t) = |Z1 | = ,
Φ(t, 0) Φ(t, u)Φ(t, v) hRe(Π
hRe U i (Ψ(t,u)+Ψ(t,v)−2Ψ(t,0))),yi
e
,
r2 (u, v, y, t) = |Z2 | = Φ(t, 0)2 Φ(t, u + v)
α1 (u, v, y, t) = arg(Z1 ) = arg
Φ(t, 0)
+ hIm(ΠhRe U i Ψ(t, u + v)), yi,
Φ(t, u)Φ(t, v)
α2 (u, v, y, t) = arg(Z2 ) = arg
Φ(t, 0)2
+ hIm(ΠhRe U i (Ψ(t, u) + Ψ(t, v)), yi.
Using (1.17) and the same arguments as in Keller-Ressel et al. [2010, Lemma
3.1], we then obtain
2
i(α1 +hβ1 ,ΠhRe U i⊥ xi)
i(α2 +hβ2 ,ΠhRe U i⊥ xi) 1 ≥ r1 e
− r2 e
= r12 + r22 − 2r1 r2 cos(α1 − α2 + hβ1 − β2 , ΠhRe U i⊥ xi)
≥ 2r1 r2 (1 − cos(α1 − α2 + hβ1 − β2 , ΠhRe U i⊥ xi)),
whence
r1 (u, v, y, t)r2 (u, v, y, t)
1
×(1−cos(α1 (u, v, y, t)−α2 (u, v, y, t)+hβ1 (u, v, t)−β2 (u, v, t), ΠhRe U i⊥ xi)) ≤ .
2
(1.18)
18
Chapter 1. Affine Processes on General State Spaces
Define now
(
R(t, y) = sup ρ ≥ 0 | r1 (u, v, y, t)r2 (u, v, y, t) >
3
for u, v ∈ ΠhRe U i⊥ U
4
)
with kImuk ≤ ρ and kImvk ≤ ρ .
Note that R(t, y) > 0 for all (t, y) ∈ R+ × ΠhRe U i D, which follows from the
fact that r1 (0, 0, y, t) = r2 (0, 0, y, t) = 1 and the continuity of the functions
(u, v) 7→ r1 (u, v, y, t)r2 (u, v, y, t). Continuity of (t, y) 7→ r1 (u, v, y, t)r2 (u, v, y, t)
also implies that (t, y) 7→ R(t, y) is continuous. Set now R(t) := inf k R(t, yk )
where yk = ΠhRe U i xk . Then (1.13) implies that R(t) > 0 for all t ≥ 0. Let
now t be fixed and g1 given by (1.15). Suppose that
hβ1 (u∗ , v ∗ , t) − β2 (u∗ , v ∗ , t), g1 i =
6 0
for some u∗ , v ∗ ∈ ΠhRe U i⊥ U with kImu∗ k < R(t) and kImv ∗ k < R(t). Then
due to the continuity of β1 and β2 , there exists some δ > 0 such that for all
u, v in a neighborhood Oδ of (u∗ , v ∗ ) defined by
n
Oδ = (u, v) ∈ (ΠhRe U i⊥ U)2 | kIm(u − u∗ )k < δ, kIm(v − v ∗ )k < δ and
o
kImuk < R(t), kImvk < R(t) ,
we also have hβ1 (u, v, t) − β2 (u, v, t), g1 i 6= 0. Moreover, there exist some
(u, v) ∈ Oδ and some k ∈ N such that
1
cos(α1 (u, v, yk , t) − α2 (u, v, yk , t) + hβ1 (u, v, t) − β2 (u, v, t), ΠhRe U i⊥ xk i) ≤ ,
3
(1.19)
since yk stays in a bounded set and ΠhRe U i⊥ xk explodes with highest divergence
rate in direction g1 . However, inequality (1.19) now implies that
r1 (u, v, yk , t)r2 (u, v, yk , t)
× 1 − cos α1 (u, v, yk , t) − α2 (u, v, yk , t)
1
+ hβ1 (u, v, t) − β2 (u, v, t), ΠhRe U i⊥ xk i > ,
2
which contradicts (1.18). Since g1 corresponds to the direction of the highest
divergence rate, we thus conclude that
hβ1 (u, v, t) − β2 (u, v, t), g1 i = Im(hΨ(t, u + v) − Ψ(t, u) − Ψ(t, v), g1 i) = 0
1.2. Càdlàg Version
19
for all u, v with kImuk < R(t) and kImvk < R(t). Continuity of u 7→ Ψ(t, u)
therefore implies that u 7→ hΨ(t, u), g1 i is a linear function. Hence there exists
a continuous curve of (real) vectors λ1 (t) ∈ hRe Ui⊥ such that
hΨ(t, u), g1 i = hλ1 (t), ui
for all u ∈ ΠhRe U i⊥ U with kImuk < R(t).
We can now proceed inductively for the remaining directions of divergence
gi . Indeed, assume that hβ1 (u, v, t) − β2 (u, v, t), gi i = 0 for all i ≤ r − 1 and
all u, v with kImuk < R(t) and kImvk < R(t). Then reapting the above steps
allows us to conclude that hβ1 (u, v, t) − β2 (u, v, t), gr i = 0 for all u, v with
kImuk < R(t) and kImvk < R(t) as well, which yields the assertion.
Consider now the set K defined in (1.12). Since (t, u) 7→ Φ(t, u)ehΨ(t,u),xi
is jointly continuous on Qm for every m ≥ 1 with Φ(0, u) = 1 and Ψ(0, u) = u
(see Proposition 1.1.6), it follows that there exists some η > 0 such that for
all t ∈ [0, η]
inf |Φ(t, u)| > c and sup kReΨ(t, u)k2 + kImΨ(t, u)k2 < C,
u∈K
(1.20)
u∈K
with some constants c and C. By fixing these constants and some linearly
independent vectors in K as described in Lemma 1.2.2, we define
ε := min(η, δ),
(1.21)
where δ > 0 is given in Lemma 1.2.2. Moreover, let t ≥ 0 be fixed. Then we
T
the set
denote by It,ε
T
It,ε
:= (t, t + ε) ∩ T ,
(1.22)
where T is defined in (1.10).
We are now prepared to prove the following proposition, which is the main
ingredient in the existence proof of a càdlàg version of X (see the proof of
Theorem (1.2.7) below).
T
Proposition 1.2.4. Let K and It,ε
be the sets defined in (1.12) and (1.22).
Consider the function Ψ given in (1.4) with the properties of Proposition 1.1.6.
Let t ≥ 0 be fixed and consider a sequence (qk )k∈N with values in Q+ ∩ [0, t]
such that qk ↑ t. Moreover, let (xqk )k∈N be a sequence with values in D∆ ∪{∞}.
Here, ∞ corresponds to a “point at infinity” and D∆ ∪ {∞} is the one-point
compactification of D∆ .3 Then the following assertions hold:
3
If the state space D is compact, we do not adjoin {∞} and only consider a sequence
with values in D∆ .
20
Chapter 1. Affine Processes on General State Spaces
T
(i) If for all (T, u) ∈ It,ε
×K
lim NqT,u
:= lim ehΨ(T −qk ,u),xqk i
k
k→∞
k→∞
(1.23)
exists finitely valued and does not vanish, then also limk→∞ xqk exists
finitely valued.
T
(ii) If there exist some (T, u) ∈ It,ε
× K such that
lim NqT,u
:= lim ehΨ(T −qk ,u),xqk i = 0,
k
k→∞
k→∞
then we have limk→∞ kxqk k = ∞.
T be a family of sequences with values in Q+ ∩ [t, T ]
Moreover, let (qkT )k∈N,T ∈It,ε
T
T
and the additional property that for every
such that qk ↓ t for every T ∈ It,ε
T
S, T ∈ It,ε , with S < T , there exists some index N ∈ N such that, for all
S
k ≥ N , qk−N
= qkT . Then the above assertions hold true for these right limits
with qk replaced by qkT .
Remark 1.2.5. Concerning assertion (ii) of Proposition 1.2.4, note that,
e.g. in the case qk ↑ t, limk→∞ kxqk k = ∞ corresponds either to explosion or
to the possibility that there exists some index N ∈ N such that xqk = ∆ for
all k ≥ N . In the latter case we also have, due to the convention k∆k = ∞,
limk→∞ kxqk k = ∞.
T
Proof. We start by proving the first assertion (i). Let T ∈ It,ε
be fixed and
define for all u ∈ K
A(u) := lim sup hReΨ(T − qk , u), xqk i ,
k→∞
a(u) := lim inf hReΨ(T − qk , u), xqk i .
k→∞
Then there exist subsequences (xqkm ) and (xqkl ) such that4
A(u) = lim ReΨ(T − qkm , u), xqkm ,
m→∞
D
E
a(u) = lim ReΨ(T − qkl , u), xqkl .
l→∞
First note that A(u) and a(u) exist finitely valued for all u ∈ K. Indeed, if
there is some u ∈ K such that A(u) = ±∞ or a(u) = ±∞, then the limit
4
Note that these subsequences depend on u. For notational convenience we however
suppress the dependence on u.
1.2. Càdlàg Version
21
of NqT,u
does not exist, or limk→∞ NqT,u
is either 0 or +∞, which contradicts
k
k
assumption (1.23). We now define
r1 (u) = lim exp ReΨ(T − qkm , u), xqkm ,
m→∞
E
D
r2 (u) = lim exp ReΨ(T − qkl , u), xqkl ,
l→∞
ϕm (u) = ImΨ(T − qkm , u), xqkm ,
E
D
ϕl (u) = ImΨ(T − qkl , u), xqkl .
Then the limits of cos(ϕm (u)), cos(ϕl (u)), sin(ϕm (u)) and sin(ϕl (u)) necessarily exist and
r1 (u) lim cos(ϕm (u)) = r2 (u) lim cos(ϕl (u)),
m→∞
l→∞
r1 (u) lim sin(ϕm (u)) = r2 (u) lim sin(ϕl (u)).
m→∞
l→∞
This yields r1 (u) = r2 (u) for all u ∈ K, since
lim cos2 (ϕm (u)) + sin2 (ϕm (u) = lim cos2 (ϕl (u)) + sin2 (ϕl (u) = 1.
m→∞
l→∞
In particular, we have proved that limk→∞ hReΨ(T − qk , u), xqk i exists finitely
T
× K. By choosing linear independent vectors
valued for all (T, u) ∈ It,ε
(u1 , . . . , up ) ∈ K ∩ Re U, it thus follows that
lim ΠhRe U i xqk
k→∞
exists finitely valued.
Therefore it only remains to focus on ΠhRe U i⊥ xqk . From the above, we
T
know in particular that for all (T, u) ∈ It,ε
×K
D
lim e
ΠhRe U i⊥ Ψ(T −qk ,u),ΠhRe U i⊥ xqk
E
k→∞
(1.24)
exists finitely valued and does not vanish. This implies that for all (T, u) ∈
T
It,ε
×K
D
E
Im ΠhRe U i⊥ Ψ(T − qk , u), ΠhRe U i⊥ xqk = αk (T, u) + 2πzk (T, u),
(1.25)
where αk (T, u) ∈ [−π, π), α(T, u) := limk→∞ αk (T, u) exists finitely valued
and (zk (T, u))k∈N is a sequence with values in Z, which a priori does not
necessarily have a limit and/or limk→∞ zk (T, u) = ±∞.
22
Chapter 1. Affine Processes on General State Spaces
Let us first assume that
lim sup kΠhRe U i⊥ xqk k = ∞.
(1.26)
k→∞
Then we are exactly in the situation of Lemma 1.2.3 and
Pthe above limit (1.24)
– after possibly selecting a subsequence such that xqk − i gi hgi , xqk i converges
as k → ∞ – can be written as
lim e
P
D
i hλi (T −qk ),ui hgi ,xqk i+
ΠhRe U i⊥ Ψ(T −qk ,u),xqk −
P
i gi
E
hgi ,xqk i
k→∞
for all u ∈ ΠhRe U i⊥ K with kImuk < P (T ), where P (T ) is defined by P (T ) :=
inf k R(T − qk ) and R and the directions gi are given in Lemma 1.2.3. Note
that due to the strict positivity and continuity of R, P (T ) is strictly positive
T
and some set MT ∗ ⊆ {u ∈
as well. Furthermore, there P
exists some T ∗ ∈ It,ε
∗
∗
6 0} of positive finite measure
ΠhRe U i⊥ K | kImuk < P (T ), i hλi (T − t), ui =
such that
Z
E
D
P
P
ΠhRe U i⊥ Ψ(T ∗ −qk ,u),xqk − i gi hgi ,xqk i ( i hλi (T ∗ −qk ),ui hgi ,xq i)
k
e
du 6= 0.
e
lim
k→∞
MT∗
(1.27)
However, it follows from the Riemann-Lebesgue Lemma that the previous
limit is zero, whence contradicting (1.27). We therefore conclude that
lim sup kΠhRe U i⊥ xqk k < ∞.
k→∞
T
× K and N ∈ N
This in turn implies that there exists some (T ∗ , u∗ ) ∈ It,ε
such that for all k ≥ N
E
D
∗
∗
Im ΠhRe U i⊥ Ψ(T − qk , u ), ΠhRe U i⊥ xqk ∈ (−π, π).
Indeed, this follows from the fact that for every u ∈ K and η > 0 there exists
T
some T ∗ ∈ It,ε
and N ∈ N such that for all k ≥ N
kIm(ΠhRe U i⊥ Ψ(T ∗ − qk , u) − ΠhRe U i⊥ u)k ≤ η.
(1.28)
For u∗ with kIm(ΠhRe U i⊥ u∗ )k sufficiently small and k sufficiently large, we thus
have
D
E
∗
∗
ΠhRe U i⊥ Ψ(T − qk , u ), ΠhRe U i⊥ xqk ≤ (kIm(ΠhRe U i⊥ u∗ )k + kIm(ΠhRe U i⊥ Ψ(T ∗ − qk , u∗ ) − ΠhRe U i⊥ u∗ )k)
× (lim sup kΠhRe U i⊥ xqk k + 1)
k→∞
< π.
1.2. Càdlàg Version
23
Hence
E
D
lim Im ΠhRe U i⊥ Ψ(T ∗ − qk , u∗ ), ΠhRe U i⊥ xqk = α(T ∗ , u∗ ).
k→∞
(1.29)
As we can find n − p linear independent vectors up+1 , . . . , un such that (1.29)
is satisfied, we conclude using Lemma 1.2.2 that
lim ΠhRe U i⊥ xqk
k→∞
exists finitely valued. This proves assertion (i).
Concerning the second statement, observe that we have
lim ehΨ(T −qk ,u),xqk i = 0,
k→∞
(1.30)
if either explosion occurs or if xqN jumps to ∆ for some N ∈ N and xqk = ∆ for
all k ≥ N . (This happens when the corresponding process is killed.) Indeed,
since (1.30) is equivalent to limk→∞ ehReΨ(T −qk ,u),xqk i = 0 and as kΨ(T − t, u)k
T
is bounded on K due to the definition of It,ε
, we necessarily have
lim kxqk k = ∞.
k→∞
In the case of a jump to ∆, this is implied by the conventions k∆k = ∞ and
f (∆) = 0 for any other function.
Similar arguments yield the assertion concerning right limits.
Using Proposition 1.2.1 and Proposition 1.2.4 above, we are now prepared
to prove Theorem 1.2.7 below, which asserts the existence of a càdlàg version
of X. Before stating this result, let us recall the notion of the (usual) augmentation of (Ft0 ) with respect to Px , which guarantees the càdlàg version to
be adapted.
Definition 1.2.6 (Usual augmentation). We denote by F x the completion
of F 0 with respect to Px . A sub-σ-algebra G ⊂ F x is called augmented with
respect to Px if G contains all Px -null-sets in F x . The augmentation of Ft0 with
respect to Px is denoted by Ftx , that is, Ftx = σ(Ft0 , N (F x )), where N (F x )
denotes all Px -null-sets in F x .
Theorem 1.2.7. Let X be an affine process relative to (Ft0 ). Then there
e such that, for each x ∈ D∆ , X
e is a Px -version of X, which
exists a process X
is càdlàg in D∆ ∪ {∞} (in D∆ respectively if D is compact) and an affine
process relative to (Ftx ). As before, ∞ corresponds to a “point at infinity” and
D∆ ∪ {∞} is the one-point compactification of D∆ , if D is not compact.
24
Chapter 1. Affine Processes on General State Spaces
e whose
Remark 1.2.8. We here establish the existence of a càdlàg version X
sample paths may take ∞ as left limiting value if D is not compact. A priori,
es− (ω) with ∆, whenever kX
es− (ω)k = ∞. Indeed, X
et (ω)
we cannot identify X
might become finitely valued for some t ≥ s. This issue is clarified in Theoet k = ∞ for all t ≥ s and all
rem 1.2.10 below, where we prove that Px -a.s. kX
es− k = ∞. In particular, this allows us to identify ∞ with ∆.
s > 0 if kX
es = ∆, which happens when the process is killed, AssumpIn the case X
et = ∆ for all t ≥ s and all s > 0.
tion (1.1) guarantees that X
e 5 , where Px [Ω]
e =
Proof. It follows from Proposition 1.2.1 that for every ω ∈ Ω
1,
t 7→ MtT,u (ω) := Φ(T − t, u)ehΨ(T −t,u),Xt (ω)i , t ∈ [0, T ],
is the restriction to Q+ ∩ [0, T ] of a càdlàg function for all (T, u) ∈ T × V.
e T and V are defined in (1.9), (1.10) and (1.11). Hence, for every
Here, Ω,
e and all (T, u) ∈ T × V, the limits
ω∈Ω
lim MqT,u (ω),
q∈Q+
q↑t
lim MqT,u (ω)
q∈Q+
q↓t
(1.31)
exist finitely valued for all t ∈ [0, T ].
Let us now show that the same holds true for X. For notational conT
venience we first focus on left limits. Consider the sets K and It,ε
defined
in (1.12) and (1.22) and let t ≥ 0 be fixed. Take some sequence (qk )k∈N , as
specified in Proposition 1.2.4, such that qk ↑ t. Then there exists some N ∈ N
T
× K, Φ(T − qk , u) 6= 0. This is a
such that, for all k ≥ N and (T, u) ∈ It,ε
consequence of the definition of ε (see (1.21)). Thus we can divide MqT,u
(ω)
k
T
by Φ(T − qk , u) for all k ≥ N and (T, u) ∈ It,ε
× K. By the continuity of
e the limit
t 7→ Φ(t, u) and (1.31), it follows that, for every ω ∈ Ω,
lim NqT,u
(ω) := lim ehΨ(T −qk ,u),Xqk (ω)i
k
k→∞
k→∞
T
exists finitely valued for all (T, u) ∈ It,ε
× K. From Proposition 1.2.4 we thus
e the limit
deduce that, for every ω ∈ Ω,
lim Xqk (ω)
k→∞
exists either finitely valued or limk→∞ kXqk (ω)k = ∞. Using similar arguments yields the same assertion for right limits. Hence we can conclude that
Px -a.s.
et = lim Xq
X
q∈Q+
q↓t
5
e ∈ F x.
Note that due to the measurable projection theorem, Ω
(1.32)
1.2. Càdlàg Version
25
exists for all t ≥ 0 and defines a càdlàg function in t.
et (ω) exists for every
Let now Ω0 be the set of ω ∈ Ω for which the limit X
t and defines a càdlàg function in t. Then, as a consequence of Rogers and
Williams [1994, Theorem II.62.7, Corollary II.62.12], Ω0 ∈ F 0 and Px [Ω0 ] = 1
et (ω) = ∆ for all t. Then X
e is a càdlàg
for all x ∈ D∆ . For ω ∈ Ω \ Ω0 , we set X
et is F 0 -measurable for every t ≥ 0. Since X is assumed to be
process and X
stochastically continuous, we have Xs → Xt in probability as s → t. Using
the fact that convergence in probability implies almost sure convergence along
a subsequence, we have


Px  lim Xq = Xt  = 1.
q∈Q+
q↓t
(1.33)
et , the limit in (1.33) is equal to X
et on Ω0 . Hence, for
By our definition of X
et = Xt ] for each t, implying that X
e is a version of
all x ∈ D∆ , we have Px [X
X. This then also yields
h
i
e
Ex ehu,Xt i = Ex ehu,Xt i
et ∈ Ftx for each
and augmentation of (Ft0 ) with respect to Px ensures that X
e is an affine process with respect to (F x ).
t. We therefore conclude that X
t
If D is non-compact, the càdlàg version (1.32) on D∆ ∪ {∞}, still denoted
by X, can be realized on the space Ω0 := D0 (D∆ ∪ {∞}) of càdlàg paths
ω : R+ → D∆ ∪ {∞} with ω(t) = ∆ for t ≥ s, whenever ω(s) = ∆. However,
we still have to prove that we can identify ∞ with ∆, as mentioned in Remark 1.2.8. In other words, we have to show that kω(t)k = ∞ for all t ≥ s if
explosion occurs for some s > 0, that is, kω(s−)k = ∞. This is stated in the
Theorem 1.2.10 below. For its proof let us introduce the following notations:
Due to the convention k∆k = ∞, we can define the explosion time by
(see Cheridito, Filipović, and Yor [2005] for a similar definition)
Texpl :=
T∆ , if Tk0 < T∆ for all k,
∞, if Tk0 = T∆ for some k,
where the stopping times T∆ and Tk0 are given by
T∆ := inf{t > 0 | kXt− k = ∞ or kXt k = ∞},
Tk0 := inf{t | kXt− k ≥ k or kXt k ≥ k}, k ≥ 1.
26
Chapter 1. Affine Processes on General State Spaces
Moreover, we denote by relint(C) the relative interior of a set C defined
by
relint(C) = {x ∈ C | B(x, r) ∩ aff(C) ⊆ C for some r > 0},
where aff(C) denotes the affine hull of C.
Lemma 1.2.9. Let X be an affine process with càdlàg paths in D∆ ∪ {∞}
and let x ∈ D be fixed. If
Px [Texpl < ∞] > 0,
(1.34)
then relint(Re U) 6= ∅ and we have Px -a.s.
lim ehu,Xt i = 0
t↑Texpl
for all u ∈ relint(Re U).
Proof. Let us first establish that under Assumption (1.34), relint Re U 6= ∅.
To this end, we denote by Ωexpl the set
Ωexpl = {ω ∈ Ω0 | Texpl (ω) < ∞}.
Then it follows from Proposition 1.2.1 and 1.2.4 that, for Px -almost every
ω ∈ Ωexpl , there exist some (T (ω), v(ω)) ∈ (Texpl (ω), ∞) × i V such that
Φ(T (ω) − t, v(ω)) 6= 0
lim
t↑Texpl (ω)
and
lim
t↑Texpl (ω)
T (ω),v(ω)
Nt
(ω) =
lim
ehΨ(T (ω)−t,v(ω)),Xt (ω)i = 0.
(1.35)
t↑Texpl (ω)
This implies that
lim hReΨ(T (ω) − t, v(ω)), Xt (ω)i = −∞,
(1.36)
t↑Texpl (ω)
and in particular that U 3 ReΨ(T (ω) − Texpl (ω), v(ω)) 6= 0, which proves the
claim, since Re U ⊆ relint Re U.
Furthermore, by (1.36) we have
lim
t↑Texpl (ω)
kΠhRe U i (Xt (ω))k = ∞.
(1.37)
1.2. Càdlàg Version
27
Define now the vector space W by
W = Re U ∩ (−Re U).
By the definition of U, |hw, Xt (ω)i| is bounded for all t ≤ Texpl (ω) and w ∈ W ,
which implies that
lim
t↑Texpl (ω)
kΠ1 (Xt (ω))k = ∞,
(1.38)
where Π1 denotes the projection on the orthogonal complement of W in
hRe Ui. As Π1 (Re U) is a proper convex cone6 (see, e.g., Bruns and Gubeladze [2009, Proposition 1.18]), we thus have for all u ∈ relint(Π1 (Re U))
lim hu, Xt (ω)i = −∞.
t↑Texpl (ω)
Writing relint(Re U) as
relint(Re U) = W + relint(Π1 (Re U)),
then yields the assertion, since |hw, Xt (ω)i| is bounded for all t ≤ Texpl (ω)
and w ∈ W .
Theorem 1.2.10. Let X be an affine process with càdlàg paths in D∆ ∪ {∞}.
Then, for every x ∈ D, the following assertion holds Px -a.s.: If
kXs− k = ∞,
(1.39)
then kXt k = ∞ for all t ≥ s and s ≥ 0. Identifying ∞ with ∆, then yields
Xt = ∆ for all t ≥ s.
Proof. Let x ∈ D be fixed and let u ∈ relint(Re U). Note that by Lemma 1.2.9
relint(Re U) 6= ∅ and that Φ(t, u) and Ψ(t, u) are real-valued functions with
values in R++ and Re U, respectively. Take now some T > 0 and δ > 0 such
that
Px [T − δ < Texpl ≤ T ] > 0,
and Ψ(t, u) ∈ relint(Re U) for all t < δ. Consider the martingale
MtT,u = Φ(T − t, u)ehΨ(T −t,u),Xt i ,
t ≤ T,
which is clearly nonnegative and has càdlàg paths. Moreover, by the choice of
δ, it follows from Lemma 1.2.9 and the conventions k∆k = ∞ and f (∆) = 0
for any other function that Px -a.s.
T,u
Ms−
= 0,
6
s ∈ (T − δ, T ],
A cone is called proper if K ∩ (−K) = {0} (see also Chapter 2).
(1.40)
28
Chapter 1. Affine Processes on General State Spaces
if and only if kXs− k = ∞ for s ∈ (T − δ, T ]. We thus conclude using Rogers
and Williams [1994, Theorem II.78.1] that Px -a.s. MtT,u = 0 for all t ≥ s,
which in turn implies that kXt k = ∞ for all t ≥ s. This allows us to identify
∞ with ∆ and we obtain Xt = ∆ for all t ≥ s. Since T was chosen arbitrarily,
the assertion follows.
Combining Theorem 1.2.7 and Theorem 1.2.10 and using Assumption (1.1),
we thus obtain the following statement:
Corollary 1.2.11. Let X be an affine process relative to (Ft0 ). Then there
e such that, for each x ∈ D∆ , X
e is a Px -version of X,
exists a process X
x
which is an affine process relative to (Ft ), whose paths are càdlàg and satisfy
et = ∆ for t ≥ s, whenever kX
es− k = ∞ or kX
es k = ∞.
Px -a.s. X
Remark 1.2.12. We will henceforth always assume that we are using the
càdlàg version of an affine process, given in Corollary 1.2.11, which we still
denote by X. Under this assumption X can now be realized on the space
Ω = D(D∆ ) of càdlàg paths ω : R+ → D∆ with ω(t) = ∆ for t ≥ s, whenever
kω(s−)k = ∞ or kω(s)k = ∞. The canonical realization of an affine process
X is then defined by Xt (ω) = ω(t).
1.3
Right-Continuity of the Filtration
Using the existence of a right-continuous version of an affine process, we can
now show that (Ftx ), that is, the augmentation of (Ft0 ) with respect to Px , is
right-continuous.
Theorem 1.3.1. Let x ∈ D be fixed and let X be an affine process relative to
(Ftx ) with càdlàg paths. Then (Ftx ) is right-continuous.
Proof. We adapt the proof of Protter [2005, Theorem I.4.31] to ourTsetting.
x
x
x
We have to show that for every t ≥ 0, Ft+
= Ftx , where Ft+
T = xs>t Ft .
x
Since the filtration is increasing, it suffices to show that Ft = n≥1 Ft+ 1 . In
n
particular, we only need to prove that
i
h
h
i
x
Ex ehu1 ,Xt1 i+···+huk ,Xtk i Ftx = Ex ehu1 ,Xt1 i+···+huk ,Xtn i Ft+
(1.41)
for all (t1 , . . . , tk ) and all (u1 , . . . , uk ) with ti ∈ R+ and ui ∈ U, since this
x
x
implies Ex [Z|Ftx ] = Ex [Z|Ft+
] for every bounded Z ∈ F x . As both Ft+
and
x
x
x
x
Ft contain the nullsets N (F ), this then already yields Ft+ = Ft for all
t ≥ 0.
1.3. Right-Continuity of the Filtration
29
In order to prove (1.41), let t ≥ 0 be fixed and take first t1 ≤ t2 · · · ≤ tk ≤ t.
Then we have for all (u1 , . . . , uk )
i
h
h
i
x
Ex ehu1 ,Xt1 i+···+huk ,Xtk i Ftx = Ex ehu1 ,Xt1 i+···+huk ,Xtk i Ft+
= ehu1 ,Xt1 i+···+huk ,Xtk i .
In the case tk > tk−1 · · · > t1 > t, we give the proof for k = 2 for notational
convenience. Let t2 > t1 > t and fix u1 , u2 ∈ U. Then we have by the affine
property
i
h
i
h
hu1 ,Xt1 i+hu2 ,Xt2 i x
hu1 ,Xt1 i+hu2 ,Xt2 i x
Ex e
Ft+ = lim Ex e
Fs
s↓t
h h
i i
hu1 ,Xt1 i+hu2 ,Xt2 i x x
= lim Ex Ex e
Ft1 Fs
s↓t
i
h
hu1 +Ψ(t2 −t1 ,u2 ),Xt1 i x
= Φ(t2 − t1 , u2 ) lim Ex e
Fs .
s↓t
If Φ(t2 −t1 , u2 ) = 0, it follows by the same step that Ex [ehu1 ,Xt1 i+hu2 ,Xt2 i | Ftx ] =
0, too. Otherwise, we have by Proposition 1.1.6 (ii), Ψ(t2 − t1 , u2 ) ∈ U, and
by the definition of U also u1 + Ψ(t2 − t1 , u2 ) ∈ U. Hence, again by the affine
property and right-continuity of t 7→ Xt (ω), the above becomes
h
i
hu1 ,Xt1 i+hu2 ,Xt2 i x
Ex e
Ft+
= Φ(t2 − t1 , u2 ) lim Φ(t1 − s, u1 + Ψ(t2 − t1 , u2 ))ehΨ(t1 −s,u1 +Ψ(t2 −t1 ,u2 )),Xs i
s↓t
= Φ(t2 − t1 , u2 )Φ(t1 − t, u1 + Ψ(t2 − t1 , u2 ))ehΨ(t1 −t,u1 +Ψ(t2 −t1 ,u2 )),Xt i
i
h
= Ex ehu1 ,Xt1 i+hu2 ,Xt2 i Ftx .
x
This yields (1.41) and by the above arguments we conclude that Ft+
= Ftx
for all t ≥ 0.
Remark 1.3.2. A consequence of Theorem 1.3.1 is that (Ω, Ft , (Ftx ), Px ) satisfies the usual conditions, since
(i) F x is Px -complete,
(ii) F0x contains all Px -null-sets in F x ,
(iii) (Ftx ) is right-continuous.
Let us now set
F :=
\
x∈D∆
F x,
Ft :=
\
x∈D∆
Ftx .
(1.42)
30
Chapter 1. Affine Processes on General State Spaces
Then (Ω, F, (Ft ), Px ) does not necessarily satisfy the usual conditions, but
Ft = Ft+ still holds true. Moreover, it follows e.g. from Revuz and Yor [1999,
Proposition III.2.12, III.2.14] that, for each t, Xt is Ft -measurable and a
Markov process relative to (Ft ).
Unless otherwise mentioned, we henceforth always consider affine processes on the filtered space (Ω, F, (Ft )), where Ω = D(D∆ ), as described in
Remark 1.2.12, and F, Ft are given by (1.42). Notice that these assumptions
on the probability space correspond to the standard setting considered for Feller
processes (compare Rogers and Williams [1994, Definition III.7.16, III.9.2]).
Similar as in the case of Feller processes, we can now formulate and prove
the strong Markov property for affine processes using the above setting and
in particular the right-continuity of the sample paths.
Theorem 1.3.3. Let X be an affine process and let T be a (Ft )-stopping time.
Then for each bounded Borel measurable function f and s ≥ 0
Ex [f (XT +s )|FT ] = EXT [f (Xs )] ,
Px -a.s.
Proof. This result can be shown by the same arguments used to prove the
strong Markov property of Feller processes (see, e.g., Rogers and Williams
[1994, Theorem 8.3, Theorem 9.4]), namely by using a dyadic approximation
of the stopping time T and applying the Markov property. Instead of using
C0 -functions and the Feller property, we here consider the family of functions
{x 7→ ehu,xi | u ∈ i V } and the affine property, which asserts in particular that
x 7→ Ex ehu,Xt i = Pt ehu,xi = Φ(t, u)ehΨ(t,u),xi
is continuous. This together with the right-continuity of paths then implies
for every Λ ∈ FT and u ∈ i V
Ex ehu,XT +s i 1Λ = Ex Ps ehu,XT i 1Λ .
The assertion then follows by the same arguments as in Rogers and Williams
[1994, Theorem 8.3] or Chung and Walsh [2005, Theorem 2.3.1].
1.4
Semimartingale Property
We shall now relate affine processes to semimartingales, where, for every
x ∈ D, semimartingales are understood with respect to the filtered probability space (Ω, F, (Ft ), Px ) defined above. By convention, we call X a semimartingale if X1[0,T∆ ) is a semimartingale, where – as a consequence of Theorem 1.2.10 and Corollary 1.2.11 – we can now define the lifetime T∆ by
T∆ (ω) = inf{t > 0 | Xt (ω) = ∆}.
(1.43)
1.4. Semimartingale Property
31
Let us start with the following definition for general Markov processes
(compare Çinlar, Jacod, Protter, and Sharpe [1980, Definition 7.1]):
Definition 1.4.1 (Extendend Generator). An operator G with domain DG is
called extended generator for a Markov process X (relative to some filtration
(Ft )) if DG consists of those Borel measurable functions f : D∆ → C for which
there exists a function Gf such that the process
Z t
Gf (Xs− )ds
f (Xt ) − f (x) −
0
is a well-defined (Ft , Px )-local martingale for every x ∈ D∆ .
In the following lemma we consider a particular class of functions for which
it is possible to state the form of the extended generator for a Markov process
in terms of its semigroup.
Lemma 1.4.2. Let X be a D∆ -valued Markov process relative to some filtration (Ft ). Suppose that u ∈ U and η > 0. Consider the function
Z
1 η
Ps ehu,xi ds.
gu,η : D → C, x 7→ gu,η (x) :=
η 0
Then, for every x ∈ D,
Mtu
Z
:= gu,η (Xt ) − gu,η (X0 ) −
0
t
1
Pη ehu,Xs− i − ehu,Xs− i ds
η
is a (complex-valued) (Ft , Px )-martingale and thus gu,η (X) is a (complexvalued) special semimartingale.
Proof. Since gu,η and Pη ehu,·i − ehu,·i are bounded, Mtu is integrable for each t
and we have
Ex [Mtu |Fr ]
= Mru .
t
1
hu,Xs− i
hu,Xs− i
Pη e
−e
ds Fr
=
+ Ex gu,η (Xt ) − gu,η (Xr ) −
r η
Z t−r
1
u
hu,Xs− i
hu,Xs− i
= Mr + EXr gu,η (Xt−r ) − gu,η (X0 ) −
Pη e
−e
ds
η
0
Z
Z
1 t−r+η
1 η
u
hu,Xr i
= Mr +
Ps e
ds −
Ps ehu,Xr i ds
η t−r
η 0
Z
Z
1 t−r+η
1 t−r
hu,Xr i
−
Ps e
ds +
Ps ehu,Xr i ds
η η
η 0
Mru
Z
32
Chapter 1. Affine Processes on General State Spaces
Hence M u is a (Ft , Px )-martingale and thus gu,η (X) a special semimartingale,
since it is the sum of a martingale and a predictable finite variation process.
Remark 1.4.3. Lemma 1.4.2 asserts that the
extended generator applied to
gu,η is given by Ggu,η (x) = η1 Pη ehu,xi − ehu,xi . Note that for general Markov
processes and even for affine processes we do not know whether the “pointwise”
infinitesimal generator applied to
Z
1 η
hu,xi
Ps ehu,xi ds,
e
= lim gu,η = lim
η→0
η→0 η 0
that is,
1
Pη ehu,xi − ehu,xi ,
η→0 η
lim
is well-defined or not.7 For this reason we consider the family of functions
{x 7→ gu,η (x) | u ∈ U, η > 0}, which exhibits in the case of affine processes similar properties as {x 7→ ehu,xi | u ∈ U} (see Remark 1.4.6 (ii) and
Lemma 1.4.7 below). These properties are introduced in the following definitions (compare Çinlar et al. [1980, Definition 7.7, 7.8]).
Definition 1.4.4 (Full Class). A class C of Borel measurable functions from
D to C is said to be a full class if, for all r ∈ N, there exists a finite family
{f1 , . . . , fN } ∈ C and a function h ∈ C 2 (CN , D) such that
x = h(f1 (x), . . . , fN (x))
(1.44)
for all x ∈ D with kxk ≤ r.
Definition 1.4.5 (Complete Class). Let β ∈ V , γ ∈ S+ (V ), where S+ (V ) denotes the positive semidefinite matrices over V , and let F be a positive measure
on V , which integrates (kξk2 ∧ 1), satisfies F ({0}) = 0 and x + supp(F ) ⊆ D∆
for all x ∈ D. Moreover, let χ : V → V denote some truncation function, that
is, χ is bounded and satisfies χ(ξ) = ξ in a neighborhood of 0. A countable
subset of functions Ce ⊂ Cb2 (D) is called complete if, for every x ∈ D, the
countable collection of numbers
κ(f (x)) = hβ, ∇f (x)i +
1X
γij Dij f (x)
2 i,j
Z
(f (x + ξ) − f (x) − h∇f (x), χ(ξ)i) F (dξ),
+
f ∈ Ce (1.45)
V
7
In the case of affine processes, this would be implied by the differentiability of Φ and
Ψ with respect to t, which we only prove in Section 1.5 using the results of this paragraph.
1.4. Semimartingale Property
33
completely determines β, γ and F . A class C of Borel measurable functions
from D to C is said to be complete class if it contains such a countable set.
Remark 1.4.6. (i) Note that the integral in (1.45) is well-defined for all
f ∈ Cb2 (D). This is a consequence of the integrability assumption and
the fact that x + supp(F ) is supposed to lie in D∆ for all x.
(ii) The class of functions
C ∗ := D → C, x 7→ ehu,xi u ∈ i V
(1.46)
is a full and complete class. Indeed, for every x ∈ D with kxk ≤ r, we
can find n linearly independent vectors (u1 , . . . , un ) such that
h π πi
.
Imhui , xi ∈ − ,
2 2
This implies that x is given by
x = arcsin Imehu1 ,xi , . . . , arcsin Imehun ,xi (Imu1 , . . . , Imun )−1
and proves that C ∗ is a full class. Completeness follows by the same
arguments as in Jacod and Shiryaev [2003, Lemma II.2.44].
Lemma 1.4.7. Let X be an affine process with Φ and Ψ given in (1.4).
Consider the class of functions
Z
1 η
hΨ(s,u),xi
C := D → C, x 7→ gu,η (x) :=
Φ(s, u)e
ds u ∈ i V, η > 0 .
η 0
(1.47)
Then C is a full and complete class.
Proof. Let (u1 , . . . un ) ∈ i V be n linearly independent vectors and define a
function fη : D → Cn by fη,i (x) = gui ,η (x). Then the Jacobi matrix Jfη (x) is
given by
 1 Rη

R
1 η
hΨ(s,u ),xi
hΨ(s,u ),xi
...

..
..

.
.
R
1 η
hΨ(s,u
),xi
n
Ψ1 (s, un )ds . . .
η 0 Φ(s, un )e
η
0
Φ(s, u1 )e
1
Ψ1 (s, u1 )ds
η
0
R
1 η
η
0
Φ(s, u1 )e
1
Ψn (s, u1 )ds
..
.
Φ(s, un )ehΨ(s,un ),xi Ψn (s, un )ds

.
In particular, the imaginary part of Jfη (x) satisfies
lim ImJfη (x) = (cos(Imhu1 , xi)Imu1 , . . . , cos(Imhun , xi)Imun )> ,
η→0
(1.48)
34
Chapter 1. Affine Processes on General State Spaces
Hence there exists some η > 0 such that the rows of ImJfη are linearly independent. As Imfη : D → Rn is a C ∞ (D)-function and as JImfη = ImJfη , it
follows from the inverse function theorem that, for each x0 ∈ D, there exists
some r0 > 0 such that Imfη : B(x0 , r0 ) → W has a C ∞ (W ) inverse, where
W = Imfη (B(x0 , r0 )).
Let now r ∈ N and consider x ∈ D with kxk ≤ r. By choosing the linearly
independent vectors (u1 , . . . , un ) and η > 0 appropriately, we can guarantee
that r0 ≥ kx0 k + r. Indeed, for every δ > 0, we can choose the linearly
independent vectors (u1 , . . . , un ) such that |hui , xi| < δ. Assume now without
loss of generality that 0 ∈ D and let x0 = 0. Due to (1.48) we can thus assure
that for all x ∈ B(0, r) ∩ D
−1
k lim JImf
(0) lim JImfη (x) − Ik
η
η→0
η→0
= k(Imu1 , . . . , Imun )−> (cos(Imhu1 , xi)Imu1 , . . . , cos(Imhun , xi)Imun )> − Ik
<1
and by the continuity of the matrix inverse the same holds true for η small
enough. The proof of the inverse function theorem (see, e.g., Howard [1997,
Theorem 4.2] or Lang [1993, Lemma XIV.1.3]) thus implies that r0 can be
chosen to be r. This proves that C is a full class.
Concerning completeness, note that
1
κ(gu,η (x)) =
η
Z
0
η
1
Φ(s, u)ehΨ(s,u),xi hβ, Ψ(s, u)i + hΨ(s, u)γΨ(s, u)i
2
!
Z
+
ehΨ(s,u),ξi − 1 − hΨ(s, u), χ(ξ)i F (dξ) ds. (1.49)
V
In particular, we have
lim κ(gu,η (x)) = κ(ehu,xi )
η→0
= ehu,xi
1
hβ, ui + hu, γui +
2
Z
!
ehu,ξi − 1 − hu, χ(ξ)i F (dξ) . (1.50)
V
By Jacod and Shiryaev [2003, Lemma II.2.44] or simply as a consequence
of the completeness of the class C ∗ , as defined in (1.46), the function u 7→
κ(ehu,xi ) admits a unique representation of form (1.50), that is, if κ(eh·,xi ) also
eγ
e γ=γ
satisfies (1.50) with (β,
e, Fe), then β = β,
e and F = Fe. This property
carries over to the class C. Indeed, for every x ∈ D, there exists some η > 0
e γ=γ
such that β = β,
e and F = Fe if u 7→ κ(gu,η (x)) also satisfies (1.49) with
eγ
(β,
e, Fe). This proves that C is a complete class.
1.4. Semimartingale Property
35
In order to establish the semimartingale property of X and to study its
characteristics, we need to handle explosions and killing. Similar to Cheridito
et al. [2005], we consider again the stopping times T∆ defined in (1.43) and Tk0
given by
Tk0 := inf{t | kXt− k ≥ k or kXt k ≥ k}, k ≥ 1.
By the convention k∆k = ∞, Tk0 ≤ T∆ for all k ≥ 1. As a transition to ∆
occurs either by a jump or by explosion, we additionally define the stopping
times:
T∆ , if Tk0 = T∆ for some k,
Tjump =
∞, if Tk0 < T∆ for all k,
T∆ , if Tk0 < T∆ for all k,
(1.51)
Texpl =
∞, if Tk0 = T∆ for some k,
0
Tk , if Tk0 < T∆ ,
Tk =
∞, if Tk0 = T∆ .
Note that {Tjump < ∞} ∩ {Texpl < ∞} = ∅ and limk→∞ Tk = Texpl with
Tk < Texpl on {Texpl < ∞}. Hence Texpl is predictable with announcing
sequence Tk ∧ k. In order to turn X into a semimartingale and to get explicit
expressions for the characteristics, we stop X before it explodes, which is
possible, since Texpl is predictable. Note that we cannot stop X before it is
killed, as Tjump is totally inaccessible. For this reason we shall concentrate on
the process (Xtτ ) := (Xt∧τ ), where τ is a stopping time satisfying 0 < τ <
Texpl , which exists by the above argument and the càdlàg property of X. Since
X = X T∆ , we have
Xtτ = Xt 1{t<(τ ∧T∆ )} + Xτ ∧T∆ 1{t≥(τ ∧T∆ )}
= Xt 1{t<(τ ∧Tjump )} + Xτ ∧Tjump 1{t≥(τ ∧Tjump )} ,
which implies that a transition to ∆ can only occur through a jump.
Recall that ∆ is assumed to be an arbitrary point which does not lie in
D. We can thus identify ∆ with some point in V \ D such that every Cb2 (D)function f can be extended continuously to D∆ with f (∆) = 0. Indeed,
without loss of generality we may assume that such a point exists, because
otherwise we can always embed D∆ in V × R.
Theorem 1.4.8. Let X be an affine process and let τ be a stopping time
with τ < Texpl , where Texpl is defined in (1.51). Then X1[0,T∆ ) and X τ are
semimartingales with state space D ∪ {0} and D∆ , respectively. Moreover, let
(B, C, ν) denote the characteristics of X τ relative to some truncation function
36
Chapter 1. Affine Processes on General State Spaces
χ. Then there exists a version of (B, C, ν), which is of the form
Z
t∧τ
bi (Xs− )ds,
Bt,i =
0
Z
t∧τ
cij (Xs− )ds,
Ct,ij =
(1.52)
0
ν(ω; dt, dξ) = K(Xt , dξ)1[0,τ ] dt,
where b : D → V and c : D → S+ (V ) are Borel measurable functions
and RK(x, dξ) is a positive kernel from (D, D) into (V, B(V )), which satisfies V (kξk2 ∧ 1)K(x, dξ) < ∞, K(x, {0}) = 0 and x + supp(K(x, ·)) ⊆ D∆
for all x ∈ D.
Proof. We adapt the proof of Çinlar et al. [1980, Theorem 7.9 (ii), (iii)] to our
setting. By Lemma 1.4.2,
Z
1 η
gu,η (X) =
Φ(s, u)ehΨ(s,u),Xi ds
η 0
is a semimartingale for every u ∈ U and η > 0. Since Lemma 1.4.7 asserts
that C, as defined in (1.47), is a full class, an application of Itô’s formula to the
function hi appearing in (1.44) shows that Xi coincides with a semimartingale
on each stochastic interval [0, τr [, where
τr = inf{t ≥ 0 | kXt k ≥ r} ∧ T∆ .
Since we have Px -a.s. limr→∞ τr = T∆ and since being a semimartingale is a
local property (see Jacod and Shiryaev [2003, Proposition I.4.25]), we conclude
that X1[0,T∆ ) is a semimartingale.
Let now τ denote a stopping time with τ < Texpl . Then X τ is also a semimartingale with state space D∆ , since explosion is avoided and the transition
to ∆ can only occur via killing, that is, a jump to ∆, which is incorporated in
the jump characteristic (see Cheridito et al. [2005, Section 3]).
By Çinlar et al. [1980, Theorem 6.25], one can find a version of the characteristics (B, C, ν) of X τ , which is of the form
Z
t∧τ
Bt,i =
ebs−,i dAs ,
Z0 t∧τ
Ct,ij =
0
e
cs−,ij dAs ,
e ω,t (dξ),
ν(ω; dt, dξ) = 1[0,τ ] dAt (ω)K
(1.53)
1.4. Semimartingale Property
37
where A is an additive process of finite variation, which is Px -indistinguishable
from a (Ft )-predictable process, eb and e
c are (Ft )-optional processes with values
e ω,t (dξ) is a positive kernel from (Ω ×
in V and S+ (V ), respectively, and K
R
e ω,t (dξ) < ∞, K
e ω,t ({0}) = 0
R+ , O(Ft ))8 into (V, B(V )) satisfying V (kξk2 ∧1)K
e ω,t ) ⊆ D∆ for all t ∈ [0, τ ] and Px -almost all ω. Moreover,
and Xt (ω) + supp(K
by Jacod and Shiryaev [2003, Theorem II.2.42], for every f ∈ Cb2 (D∆ ), the
process
Z t∧τ
Z
1 t∧τ X
τ
e
hbs− , ∇f (Xs− )idAs −
e
cs−,ij Dij f (Xs− )dAs
f (Xt ) − f (x) −
2 0
0
i,j
Z t∧τ Z
e ω,s− (dξ)dAs (1.54)
−
(f (Xs− + ξ) − f (Xs− ) − h∇f (Xs− ), χ(ξ)i) K
0
V
is a (Ft , Px )-local martingale and the last three terms are of finite variation.
Note here that ∆ is assumed to be an arbitrary point in V \ D such that we
can extend f ∈ Cb2 (D) continuously with f (∆) = 0. Let us denote
X
e (Xt− (ω)) := hebt− , ∇f (Xt− (ω))i − 1
Lf
e
ct−,ij Dij f (Xt− (ω))
2 i,j
Z
e ω,t− (dξ). (1.55)
−
(f (Xt− (ω) + ξ) − f (Xt− (ω)) − h∇f (Xt− (ω)), χ(ξ)i) K
V
As proved in Lemma 1.4.7, the class of functions C defined in (1.47) is
complete. Let now Ce ⊂ C be the countable set satisfying the property stated
in Definition 1.4.5 and let gη,u ∈ Ce for some u ∈ i V and η > 0. Then
Lemma 1.4.2 and Remark 1.4.1 imply that
Z t∧τ
τ
gη,u (Xt ) − gη,u (x) −
Ggη,u (Xs− )ds
0
Z t∧τ
1
τ
= gη,u (Xt ) − gη,u (x) −
Pη ehu,Xs− i − ehu,Xs− i ds (1.56)
η
0
R t∧τ
is a (Ft , Px )-martingale, while ( 0 Ggη,u (Xs− )ds) is a predictable finite variation process. Due to (1.54), (1.55) and uniqueness of the canonical decomposition of the special semimartingale gη,u (X τ ) (see Jacod and Shiryaev [2003,
Definition I.4.22, Corollary I.3.16]), we thus have
Z t∧τ
Z t∧τ
e η,u (Xs− )dAs =
Lg
Ggη,u (Xs− )ds up to an evanescent set.
0
0
(1.57)
8
Here, O(Ft ) denotes the (Ft )-optional σ-algebra.
38
Chapter 1. Affine Processes on General State Spaces
Set now
n
o
e η,u (X(t∧τ ∧T )− (ω)) = 0 for every gη,u ∈ Ce .
Λ = (ω, t) : Lg
∆
Then the characteristic property (1.45) of Ce implies that Λ is exactly the
e = 0. Hence we may replace A by 1Λc A
set where eb = 0, e
c = 0 and K
without altering (1.53), that is, we can suppose that 1Λ A = 0. This property
together with (1.57) implies that dAt dt Px -a.s. Hence we know that there
e replaced by
exists a triplet (b0 , c0 , K 0 ) such that A replaced by t and (eb, e
c, K)
0 0
0
(b , c , K ) satisfy all the conditions of (1.53). In particular, we have by Jacod
and Shiryaev [2003, Proposition II.2.9 (i)] that X τ is quasi-left continuous.
By Çinlar et al. [1980, Theorem 6.27], it thus follows that
b0t = b(Xt )1[0,τ ] ,
c0t = c(Xt )1[0,τ ] ,
0
Kω,t
(dξ) = K(Xt , dξ)1[0,τ ] ,
where the functions b, c and the kernel K have the properties stated in (1.52).
This proves the assertion.
1.5
Regularity
By means of the above derived semimartingale property, in particular the
fact that the characteristics are absolutely continuous with respect to the
Lebesgue measure, we can prove that every affine process is regular in the
following sense:
Definition 1.5.1 (Regularity). An affine process X is called regular if for
every u ∈ U the derivatives
∂Φ(t, u) ∂Ψ(t, u) R(u) =
(1.58)
F(u) =
,
∂t ∂t t=0
exist and are continuous on U
m
t=0
for every m ≥ 1.
n−m
Remark 1.5.2. In the case of the canonical state space D = Rm
,
+ × R
the derivative of φ(t, u) at t = 0 is denoted by F (u) (see Duffie et al. [2003,
Equation (3.10)] and Remark 1.1.5). Since Φ(t, u) = eφ(t,u) , we have
F(u) = ∂t Φ(t, u)|t=0 = eφ(0,u) ∂t φ(t, u)|t=0 = ∂t φ(t, u)|t=0 = F (u).
By the definition of R(u) in Duffie et al. [2003, Equation 3.11], we also have
R(u) = R(u). Hence F and R coincide with F and R, as defined in Duffie
et al. [2003].
1.5. Regularity
39
Lemma 1.5.3. Let X be an affine process. Then the functions t 7→ Φ(t, u)
and t 7→ Ψi (t, u), i ∈ {1, . . . , n}, defined in (1.4) are of finite variation for all
u ∈ U.
Proof. Due to Assumption 1.1.3, there
1 , . . . , xn )
Pexist n+1 vectors such that (xP
are linearly independent and xn+1 = ni=1 λi xi for some λ ∈ V with ni=1 λi 6=
1.
Let us now take n + 1 affine processes X 1 , . . . , X n+1 such that
Pxi [X0i = xi ] = 1
for all i ∈ {1, . . . , n + 1}. It then follows from Theorem 1.4.8 that, for every
i ∈ {1, . . . , n + 1}, X i is a semimartingale with respect to the filtered probability space (Ω, F, (Ft ), Pxi ). We can then construct a filtered probability
space (Ω0 , F 0 , (Ft0 ), P0 ), with respect to which X1 , . . . , Xn+1 are independent
semimartingales such that P0 ◦ (X i )−1 = Pxi . One possible construction is the
n+1
n+1
product probability space (Ωn+1 , ⊗n+1
i=1 F, (⊗i=1 Ft ), ⊗i=1 Pxi ).
We write yi = (1, xi )> and Y i = (1, X i )> for i ∈ {1, . . . , n + 1}. Then
the definition of xi implies that (y1 , . . . , yn+1 ) are linearly independent. Moreover, as X i exhibits càdlàg paths for all i ∈ {1, . . . , n + 1}, there exists some
stopping time δ > 0 such that, for all ω ∈ Ω0 and t ∈ [0, δ(ω)), the vectors (Yt1 (ω), . . . , Ytn+1 (ω)) are also linearly independent. Let now T > 0 and
u ∈ U be fixed and choose some 0 < ε(ω) ≤ δ(ω) such that, for all t ∈ [0, ε(ω)),
Φ(T − t, u) 6= 0.
i
Denoting the (Ft0 , P0 )-martingales Φ(T − t, u)ehΨ(T −t,u),Xt i by MtT,u,i and
choosing the right branch of the complex logarithm, we thus have for all
t ∈ [0, ε(ω))

  ln Φ(T − t, u) 

T,u,1
1 (ω)
1 (ω) −1
1 Xt,1
. . . Xt,n
ln Mt
(ω)
Ψ1 (T − t, u) 

 
..
..
..
..
.

 ...

=
.
..
.
.
.


.
n+1
n+1
(ω) . . . Xt,n
(ω)
1 Xt,1
ln MtT,u,n+1 (ω)
Ψn (T − t, u)
This implies that (Φ(s, u))s and (Ψ(s, u))s coincide on the stochastic interval
(T − ε(ω), T ] with deterministic semimartingales and are thus of finite variation. As this holds true for all T > 0, we conclude that t 7→ Φ(t, u) and
t 7→ Ψi (t, u) are of finite variation.
Using Lemma 1.5.3 and Theorem 1.4.8, we are now prepared to prove regularity of affine processes. Additionally, our proof reveals that the functions F
and R have parameterizations of Lévy-Khintchine type and that the (differential) semimartingale characteristics introduced in (1.52) depend in an affine
way on X.
40
Chapter 1. Affine Processes on General State Spaces
Theorem 1.5.4. Every affine process is regular. Moreover, the functions F
and R, as defined in (1.58), are of the form
1
F(u) = hu, bi + hu, aui − c
2
Z
ehu,ξi − 1 − hu, χ(ξ)i m(dξ),
+
u ∈ U,
V
1
hR(u), xi = hu, B(x)i + hu, A(x)ui − hγ, xi
2
Z
ehu,ξi − 1 − hu, χ(ξ)i M (x, dξ),
+
u ∈ U,
V
where χ : V → V denotes some truncation function such that χ(∆ − x) = 0
for all x ∈ D, b ∈ V , a ∈ S(V ), m is a (signed) measure, c ∈ R, γ ∈ V and
x 7→ B(x), x 7→ A(x), x 7→ M (x, dξ) are restrictions of R-linear maps on V
such that
b(x) = b + B(x),
c(x) = a + A(x),
K(x, dξ) = m(dξ) + M (x, dξ) + (c + hγ, xi)δ(∆−x) (dξ).
Here, the left hand side corresponds to the (differential) semimartingale characteristics introduced in (1.52).
Furthermore, on the set Q = {(t, u) ∈ R+ × U | Φ(s, u) 6= 0, for all s ∈
[0, t]}, the functions Φ and Ψ satisfy the ordinary differential equations
∂t Φ(t, u) = Φ(t, u)F(Ψ(t, u)),
∂t Ψ(t, u) = R(Ψ(t, u)),
Φ(0, u) = 1,
Ψ(0, u) = u ∈ U.
(1.59)
(1.60)
Remark 1.5.5. Recall that without loss of generality we identify ∆ with some
point in V \ D such that every f ∈ Cb2 (D) can be extended continuously to D∆
with f (∆) = 0.
Proof. Let m ≥ 1 and u ∈ U m be fixed and choose Tu > 0 such that Φ(Tu −
t, u) 6= 0 for all t ∈ [0, Tu ]. As t 7→ Φ(t, u) and t 7→ Ψ(t, u) are of finite
variation by Lemma 1.5.3, their derivatives with respect to t exist almost
everywhere and we can write
Z t
Φ(Tu − t, u) − Φ(Tu , u) =
−dΦ(Tu − s, u),
0
Z t
Ψi (Tu − t, u) − Ψi (Tu , u) =
−dΨi (Tu − s, u),
0
1.5. Regularity
41
for i ∈ {1, . . . , n}. Moreover, by the semiflow property of Φ and Ψ (see
Proposition 1.1.6 (iv)), differentiability of Φ(t, u) and Ψ(t, u) with respect to
t at some Tu ≥ ε > 0 implies that the derivatives ∂t |t=0 Ψ(t, Ψ(ε, u)) and
∂t |t=0 Φ(t, Ψ(ε, u)) exist as well. Let now (εk )k∈N denote a sequence of points
where Φ(t, u) and Ψ(t, u) are differentiable such that limk→∞ εk = 0. Then
there exists a sequence (uk )k∈N given by
uk = Ψ(εk , u) ∈ U with lim uk = u
(1.61)
k→∞
such that the derivatives
∂t |t=0 Ψ(t, uk ),
∂t |t=0 Φ(t, uk )
(1.62)
exist for every k ∈ N. Moreover, since |Ex [exp(hu, Xεk i)]| < m, there exists
some constant M such that uk ∈ U M for all k ∈ N.
Furthermore, due to Theorem 1.4.8, the canonical semimartingale representation of X τ (see Jacod and Shiryaev [2003, Theorem II.2.34]), where τ is
a stopping time with τ < Texpl , is given by
Xtτ
t∧τ
Z
=x+
b(Xs− )ds +
Ntτ
0
Z
t∧τ
Z
+
0
τ
(ξ − χ(ξ))µX (ω; ds, dξ),
V
τ
where µX is the random measure associated with the jumps of X τ and N τ
is a local martingale, namely the sum of the continuous martingale part and
the purely discontinuous one, that is,
Z
0
t∧τ
Z
τ
χ(ξ)(µX (ω; ds, dξ) − K(Xs− , dξ)ds).
V
Tu ,u
Applying Itô’s formula (relative to the measure Px ) to the martingale Mt∧τ
=
42
Chapter 1. Affine Processes on General State Spaces
Φ(Tu − (t ∧ τ ), u)ehΨ(Tu −(t∧τ ),u),Xt∧τ i , we obtain
Z t∧τ
−dΦ(Tu − s, u)
Tu ,u
Tu ,u
Tu ,u
Ms−
Mt∧τ = M0 +
+ h−dΨ(Tu − s, u), Xs− i
Φ(Tu − s, u)
0
Z t∧τ
Tu ,u
Ms−
hΨ(Tu − s, u), b(Xs− )i
+
0
1
hΨ(Tu − s, u), c(Xs− )Ψ(Tu − s, u)i
2
!
Z
ehΨ(Tu −s,u),ξi − 1 − hΨ(Tu − s, u), χ(ξ)i K(Xs− , dξ) ds
+
+
V
Z
t∧τ
Tu ,u
Ms−
hΨ(Tu − s, u), dNsτ i
Z0 t∧τ Z
Tu ,u
+
Ms−
ehΨ(Tu −s,u),ξi − 1 − hΨ(Tu − s, u), χ(ξ)i
0
V
τ
× µX (ω; ds, dξ) − K(Xs− , dξ)ds .
+
As the last two terms are local martingales and as the rest is of finite variation,
we thus have, for almost all t ∈ [0, Tu ∧ τ ], Px -a.s. for every x ∈ D,
dΦ(Tu − t, u)
+ hdΨ(Tu − t, u), Xt− i
Φ(Tu − t, u)
1
= hΨ(Tu − t, u), b(Xt− )i dt + hΨ(Tu − t, u), c(Xt− )Ψ(Tu − t, u)i dt
2
Z
+
ehΨ(Tu −t,u),ξi − 1 − hΨ(Tu − t, u), χ(ξ)i K(Xt− , dξ)dt.
V
(1.63)
Note in particular that due to x + supp(K(x, ·)) ⊆ D∆ for every x ∈ D, the
above integral is well-defined. By setting t = Tu on a set of positive measure
with Px [τ ≥ Tu ] and letting Tu → 0, we obtain due to (1.62) for all k ∈ N and
x∈D
∂t |t=0 Φ(t, uk ) + h∂t |t=0 Ψ(t, uk ), xi
Z
1
= huk , b(x)i dt + huk , c(x)uk i dt +
ehuk ,ξi − 1 − huk , χ(ξ)i K(x, dξ)dt,
2
V
(1.64)
where (uk ) is given by (1.61). Since the right hand side is continuous in uk ,
which is again a consequence of the support properties of K(x, ·) and the
fact that uk ∈ U M for all k ∈ N, the limit for uk → u of the left hand side
1.5. Regularity
43
exists as well. By the affine independence of the n + 1 elements in D, the
coefficients ∂t |t=0 Φ(t, uk ) and ∂t |t=0 Ψ(t, uk ) converge for uk → u, whence the
limit is affine, too. Since u was arbitrary, it follows that
1
hu, b(x)i dt + hu, c(x)ui dt +
2
Z
ehu,ξi − 1 − hu, χ(ξ)i K(x, dξ)dt
V
is an affine function in x for all u ∈ U.
By uniqueness of the Lévy-Khintchine representation and the assumption
that D contains n+1 affinely independent elements, this implies that x 7→ b(x),
x 7→ c(x) and x 7→ K(x, dξ) are affine functions in the following sense:
b(x) = b + B(x),
c(x) = a + A(x),
K(x, dξ) = m(dξ) + M (x, dξ) + (c + hγ, xi)δ(∆−x) (dξ),
where b ∈ V , a ∈ S(V ), m a (signed) measure, c ∈ R, γ ∈ V and x 7→ B(x),
x 7→ A(x), x 7→ M (x, dξ) are restriction of R-linear maps on V . Indeed,
c + hγ, xi corresponds to the killing rate of the process, which is incorporated
in the jump measure. Here, we explicitly use the convention ehu,∆i = 0 and
the fact that χ(∆ − x) = 0 for all x ∈ D.
Moreover, for t small enough, we now have for all u ∈ U
t
Z
Φ(t, u) − Φ(0, u) =
1
hΨ(s, u), aΨ(s, u)i − c
2
!
hΨ(s,u),ξi
e
− 1 − hΨ(s, u), χ(ξ)i m(dξ) ds,
Φ(s, u) hΨ(s, u), bi +
0
Z
+
V
Z
t
hΨ(t, u) − Ψ(0, u), xi =
hΨ(s, u), B(x)i +
0
1
hΨ(s, u), A(x)Ψ(s, u)i
2
− hγ, xi
!
Z
+
ehΨ(s,u),ξi − 1 − hΨ(s, u), χ(ξ)i M (x, dξ) ds.
V
Note again that the properties of the support of K(x, ·) carry over to the
measures M (x, ·) and m(·) implying that the above integrals are well-defined.
Due to the continuity of t 7→ Φ(t, u) and t 7→ Ψ(t, u), we can conclude that the
derivatives of Φ and Ψ exist at 0 and are continuous on U m for every m ≥ 1,
44
Chapter 1. Affine Processes on General State Spaces
since they are given by
∂Φ(t, u) F(u) =
∂t = hu, bi +
t=0
Z
+
1
hu, aui − c
2
ehu,ξi − 1 − hu, χ(ξ)i m(dξ),
V
*
hR(u), xi =
∂Ψ(t, u) ∂t +
,x
1
hu, A(x)ui − hγ, xi
2
ehu,ξi − 1 − hu, χ(ξ)i M (x, dξ).
= hu, B(x)i +
t=0
Z
+
V
This proves the first part of the theorem.
By the regularity of X, we are now allowed to differentiate the semiflow
equations (1.5) on the set Q = {(t, u) ∈ R+ × U | Φ(s, u) 6= 0, for all s ∈ [0, t]}
with respect to s and evaluate them at s = 0. As a consequence, Φ and Ψ
satisfy (1.59) and (1.60).
Remark 1.5.6. The differential equations (1.59) and (1.60) are called generalized Riccati equations, which is due to the particular form of F and R.
Chapter 2
Affine Processes on Proper
Convex Cones
In order to be able to further characterize the functions F and R, as introduced
in (1.58), and to show that the generalized Riccati equations admit unique
global solutions, which is needed to establish existence of affine processes, we
assume in this chapter the state space D to be a closed proper convex cone.
We denote it by K to emphasize the difference to the case of a general state
space. This setting then allows us to prove existence of affine pure jump
processes, which is elaborated in Section 2.5. The question of existence of
affine diffusion processes is treated in Chapter 3, where we suppose further
structural assumptions on the state space by building on the framework of
irreducible symmetric cones. Clearly, the case Rn+ studied in Duffie et al.
[2003] is an example of a proper convex cone.1
Most proofs in this chapter are generalizations of the corresponding results on the cone of positive semidefinite matrices obtained in Cuchiero et al.
[2011a].
2.1
Definition of Cone-valued Affine Processes
Following Faraut and Korányi [1994], we call a cone proper if K ∩(−K) = {0}.
We further suppose K to be generating, that is, K contains a basis, which is
equivalent to V = K − K. The closed dual cone of K is defined by
K ∗ = {u ∈ V | hx, ui ≥ 0 for all x ∈ K}.
Note that the above assumptions imply in particular that K ∗ is also generating
and proper (see, e.g., Faraut and Korányi [1994, Proposition I.1.4]). We denote
1
Let us remark that Rn+ is actually a reducible symmetric cone (see Chapter 3 for details).
45
46
Chapter 2. Affine Processes on Proper Convex Cones
the open dual cone of K by
K̊ ∗ = {u ∈ V | hx, ui > 0 for all x ∈ K},
and by
∂K ∗ = {u ∈ V | hx, ui = 0 for some x ∈ K}
the boundary of K ∗ . Like any cone, K ∗ induces a partial and strict order
relation on V : for u, v ∈ V , we write u v if v − u ∈ K ∗ and u ≺ v if
v − u ∈ K̊ ∗ . As in Chapter 1, we adjoin to the state space K a point ∆ ∈
/K
and set K∆ = K ∪ {∆}.
Since the set U, as defined in (1.3), certainly contains −K ∗ , we can slightly
modify the definition of an affine process, by requiring the affine property only
for u ∈ −K ∗ . Thus, instead of the Fourier-Laplace transform, we here only
consider the Laplace transform of X, implying that
Ex ehu,Xt i = Φ(t, u)ehΨ(t,u),xi
is real-valued and cannot become 0 for u ∈ −K ∗ . We thus adjust the definition
of an affine process on K as follows:
Definition 2.1.1 (Cone-valued affine process). A time-homogeneous Markov
process X relative to some filtration (Ft ) and with state space K (augmented
by ∆) is called affine if
(i) it is stochastically continuous, that is, lims→t ps (x, ·) = pt (x, ·) weakly on
K for every t ≥ 0 and x ∈ K, and
(ii) its Laplace transform has exponential-affine dependence on the initial
state. This means that there exist functions φ : R+ × K ∗ → R and
ψ : R+ × K ∗ → V such that
Z
−hu,Xt i −hu,xi
Ex e
= Pt e
=
e−hu,ξi pt (x, dξ) = e−φ(t,u)−hψ(t,u),xi , (2.1)
K
for all x ∈ K and (t, u) ∈ R+ × K ∗ .
Remark 2.1.2. (i) Let us remark that the above definition and Definition 1.1.4 are equivalent in the case D = K. Indeed, if X is an affine
process with state space D = K in the sense of Definition 1.1.4, then
it is clearly also an affine process in the sense of Definition 2.1.1, since
the only difference is the restriction of U to −K ∗ .
Note that for u ∈ K ∗ we have
Φ(t, −u) = e−φ(t,u)
and
Ψ(t, −u) = −ψ(t, u).
2.1. Definition of Cone-valued Affine Processes
47
For the other direction we have to show that (2.1) can be extended to U,
where it takes the form
Ex ehu,Xt i = Φ(t, u)ehΨ(t,u),xi .
Note that U = −K ∗ + i V , since K is a cone. Following the proof
of Keller-Ressel et al. [2010, Lemma 2.5], let us define the function
g(t, u, x) = Ex ehu,Xt i = Pt ehu,xi
for (t, u, x) ∈ R+ × U × K. Due to (2.1), we have
g(t, u, x)g(t, u, y) = g(t, u, x + y)g(t, u, 0)
(2.2)
for all (t, u) ∈ R+ × −K ∗ and x, y ∈ K. Now it follows from wellknown properties of the Fourier-Laplace transform (see, e.g., Duffie et al.
[2003, Lemma A.2]) that both sides are analytic on Ů = −K̊ ∗ + i V .
Moreover, they coincide on a set of uniqueness, namely −K̊ ∗ , since K ∗
is generating. By the continuity of the Fourier-Laplace transform in u,
equality (2.2) therefore holds on all of U.
Assume now that g(t, u, 0) = 0for some
(t, u) ∈ R+ × U, then it follows
from (2.2) that g(t, u, x) = Ex ehu,Xt i = 0 for all x ∈ K. On the other
hand the set
O = {(t, u) ∈ R+ × U | g(t, u, 0) 6= 0}
is open, which is a consequence of the joint continuity of (t, u) 7→ g(t, u, x),
and we can define
g(t, u, x)
h(x) =
g(t, u, 0)
for all (t, u) ∈ R+ × O. The function h is measurable and satisfies
h(0) 6= 0 and h(x)h(y) = h(x + y) for all x, y ∈ K. Using a standard
result on measurable solutions of the Cauchy equation (see, e.g., Aczél
[1966, Section 2.2]), we conclude that there exists a unique continuous
extension of −ψ(t, −u) from −K ∗ to U such that
g(t, u, x)
= ehΨ(t,u),xi .
g(t, u, 0)
Setting Φ(t, u) = g(t, u, 0) completes the proof.
Note that the arguments allowing to extend the affine property from −K ∗
to U rely on the cone structure of K, as we need (2.2) to hold for sufficiently many x, y ∈ V .
48
Chapter 2. Affine Processes on Proper Convex Cones
(ii) Due to the equivalence of Definition 1.1.4 and Definition 2.1.1, we can
consider affine processes on proper convex cones on the filtered space
(Ω, F, (Ft )), where Ω = D(K∆ ), as described in Remark 1.2.12, and
where F, Ft are given in (1.42). This assumption however only plays a
role in Theorem 3.7.2 of Section 3.7.
Similar to Proposition 1.1.6, the following properties of φ and ψ are immediate consequences of Definition 2.1.1.
Proposition 2.1.3. Let X be an affine process on K. Then the functions φ
and ψ satisfy the following properties.
(i) φ maps R+ × K ∗ into R+ and ψ maps R+ × K ∗ into K ∗ .
(ii) φ and ψ satisfy the semiflow property, that is, for any s, t ≥ 0 and
u ∈ K ∗ we have
φ(t + s, u) = φ(t, u) + φ(s, ψ(t, u)),
ψ(t + s, u) = ψ(s, ψ(t, u)),
φ(0, u) = 0,
ψ(0, u) = u.
(2.3)
(2.4)
(iii) φ and ψ are jointly continuous on R+ × K ∗ . Furthermore, u 7→ φ(t, u)
and u 7→ ψ(t, u) are analytic on K̊ ∗ .
(iv) For any t ≥ 0 and u, v ∈ K ∗ with u v the order relations
φ(t, u) ≤ φ(t, v) and
ψ(t, u) ψ(t, v)
hold true.
Proof. The left hand side of (2.1) is clearly bounded by 1 for all x ∈ K.
Inserting first x = 0 shows that φ(t, u) can only take values in R+ . For
arbitrary x ∈ K the right hand side remains bounded only if ψ(t, u) ∈ K ∗ ,
which shows (i).
Assertion (ii) follows from the law of iterated expectations as in Proposition 1.1.6 (iv) or equivalently from the Chapman-Kolmogorov equation, that
is,
Z
−φ(t+s,u)−hψ(t+s,u),xi
e
=
e−hu,ξi pt+s (x, dξ)
ZK
Z
e
e
=
ps (x, dξ)
e−hu,ξi pt (ξ, dξ)
K
K
Z
= e−φ(t,u)
e−hψ(t,u),ξi ps (x, dξ)
K
= e−φ(t,u)−φ(s,ψ(t,u))−hψ(s,ψ(t,u)),xi .
2.2. Feller Property and Regularity
49
Taking logarithms and using the fact that K is generating, yields (ii).
Joint continuity of φ and ψ can be shown by the same arguments used to
prove Proposition 1.1.6 (i). The second assertion is a consequence of analyticity properties of the Laplace transform.
Concerning (iv), let u v, which is equivalent to hu, xi ≤ hv, xi for all
x ∈ K. Hence, for all t ≥ 0 and x ∈ K, we have
Z
Z
−φ(t,u)−hψ(t,u),xi
−hu,ξi
e
=
e
pt (x, dξ) ≥
e−hv,ξi pt (x, dξ) = e−φ(t,v)−hψ(t,v),xi ,
K
K
which yields (iv).
2.2
Feller Property and Regularity
The assumption of a cone state space allows us to prove that (Pt ) is a Feller
semigroup on C0 (K). In order to show this property, we shall mainly rely
on Lemma 2.2.1 below. In addition, this result also enables us to give an
alternative proof of Theorem 1.5.4, showing regularity on −K ∗ for every affine
process on K. Indeed, regularity for affine processes on cone state spaces
can be obtained by arguing as in Keller-Ressel et al. [2010], who obtained
n−m
the corresponding statements on the canonical state space D = Rm
+ ×R
(see Keller-Ressel et al. [2010, Theorem 4.3] and also the PhD thesis of KellerRessel [2009]). Let us remark that Theorem 1.5.4 can of course be applied
here, but since the assumption of a cone state space allows for a simpler
proof, we briefly reformulate the arguments of Keller-Ressel et al. [2010] for
our setting.
Lemma 2.2.1. Let ψ : R+ × K ∗ → V be any map satisfying ψ(0, u) = u and
the properties (i)–(iv) of Proposition 2.1.3 (regarding the function ψ). Then
ψ(t, u) ∈ K̊ ∗ for all (t, u) ∈ R+ × K̊ ∗ .
Proof. We adapt the proof of Keller-Ressel [2009, Proposition 1.10] to our
setting. Assume by contradiction that there exists some (t, u) ∈ R+ × K̊ ∗
such that ψ(t, u) ∈ ∂K ∗ . We show that in this case also ψ( 2t , u) ∈ ∂K ∗ . First
note that
t
t
t
ψ
,v ψ
,ψ
,u
= ψ(t, u)
(2.5)
2
2
2
for all v ∈ Θ := {v ∈ K ∗ : v ψ( 2t , u)} by Proposition 2.1.3 (ii) and (iv). Take
now some x 6= 0 ∈ K such that hx, ψ(t, u)i = 0. By (2.5) also hx, ψ( 2t , v)i = 0
for all v ∈ Θ. If ψ( 2t , u) ∈ K̊ ∗ , then Θ is a set with non-empty interior. By
50
Chapter 2. Affine Processes on Proper Convex Cones
analyticity of ψ, it then follows that hx, ψ( 2t , u)i = 0 for all u ∈ K̊ ∗ , and hence
ψ( 2t , u) ∈ ∂K ∗ , which is a contradiction. We conclude that ψ( 2t , u) ∈ ∂K ∗ .
Repeating these arguments yields, for each n ∈ N, the existence of an
element xn 6= 0 ∈ K, for which
t
xn , ψ
,u
= 0.
2n
Without loss of generality we may assume that kxn k = 1 for each n. Since
the unit sphere is compact in finite dimensions, there exists a subsequence nk
such that xnk → x∗ 6= 0, as k → ∞. From the continuity of the function
t 7→ ψ(t, u) and the scalar product we deduce that
t
,u
= hx∗ , ψ(0, u)i = hx∗ , ui > 0,
0 = lim xnk , ψ
k→∞
2nk
which is the desired contradiction.
It is now a direct consequence of this lemma that any affine process X on
K is a Feller process.
Theorem 2.2.2. Let X be an affine process on K. Then X is a Feller process.
Proof. By Revuz and Yor [1999, Proposition III.2.4], it suffices to show that
for all f ∈ C0 (K)
lim Pt f (x) = f (x),
for all x ∈ K,
(2.6)
Pt f ∈ C0 (K),
for all t ≥ 0.
(2.7)
t↓0
Property (2.6) is a consequence of stochastic continuity, which implies for
all f ∈ C0 (K) and x ∈ K
lim Pt f (x) = f (x).
t↓0
Concerning (2.7), it suffices to verify this property for a dense subset of
C0 (K). By a locally compact version of Stone-Weierstrass’
Theorem
(see,
n
o
−hu,xi
∗
e.g., Semadeni [1971]), the linear span of the set e
| u ∈ K̊
is dense
in C0 (K). Indeed, it is a subalgebra of C0 (K), separates points and vanishes nowhere, as all elements are strictly positive functions on K. From
Lemma 2.2.1 we can deduce that Pt e−hu,xi ∈ C0 (K) if u ∈ K̊ ∗ , since ψ(t, u) ∈
K̊ ∗ and hψ(t, u), xi > 0 for x 6= 0 implying that
Pt e−hu,xi = e−φ(t,u)−hψ(t,u),xi
goes to 0 as x → ∆, whence the assertion is proved.
2.2. Feller Property and Regularity
51
As already mentioned before, Lemma (2.2.1) now allows us to prove regularity of affine processes on K using a different approach than the one of
Section 1.5. In order to avoid confusion concerning the different notations, let
us give again a rigorous definition of regularity for an affine process on K.
Definition 2.2.3 (Regularity for cone-valued affine processes). An affine process X on K is called regular if for all u ∈ K ∗ the derivatives
∂φ(t, u) ∂ψ(t, u) F (u) =
R(u) =
(2.8)
,
∂t ∂t t=0
t=0
exist and are continuous in u.
Remark 2.2.4. Note that in the terminology of Definition 1.5.1, we here
require regularity on −K ∗ instead of U. Moreover, we have
F(−u) = ∂t Φ(t, −u)|t=0 = −e−φ(0,u) ∂t φ(t, u)|t=0 = −∂t φ(t, u)|t=0 = −F (u),
R(−u) = ∂t Ψ(t, −u)|t=0 = −∂t ψ(t, u)|t=0 = −R(u).
Let us now reformulate parts of Theorem 1.5.4 in the context of cone state
spaces:
Theorem 2.2.5. Let X be an affine process on K. Then X is regular in
the sense of Definition 2.2.3 and the functions φ and ψ satisfy the ordinary
differential equations
∂t φ(t, u) = F (ψ(t, u)),
∂t ψ(t, u) = R(ψ(t, u)),
φ(0, u) = 0,
ψ(0, u) = u ∈ K ∗ .
(2.9)
(2.10)
Proof. A proof of the above theorem can be obtained by following the lines
of Keller-Ressel et al. [2010, Proof of Theorem 4.3]. Observe first that Proposition 2.1.3 (iii) implies differentiability of φ(t, u) and ψ(t, u) in u on K̊ ∗ .
Arguing similarly as in the proof of Keller-Ressel et al. [2010, Theorem 4.3]
and using Lemma 2.2.1, yields differentiability of φ(t, u) and ψ(t, u) at t = 0
and continuity of the derivatives in u for all u ∈ K ∗ .
Similarly as in the proof of Theorem 1.5.4, we are now allowed to differentiate the semiflow equations (2.3) and (2.4) with respect to s and evaluate
them at s = 0. As a consequence, φ and ψ satisfy (2.9) and (2.10).
Remark 2.2.6. Note that in contrast to the general state space (2.9) and (2.10)
hold for all (t, u) ∈ R+ × K ∗ .
52
2.3
Chapter 2. Affine Processes on Proper Convex Cones
Necessary Conditions
In this section, we focus on the specific form of the functions F and R, defined in (2.8). As already proved in Theorem 1.5.4 for the case of general
state spaces, F and R have parameterizations of Lévy-Khintchine type. In
Proposition 2.3.2 below, we give an alternative proof of this result, which subsequently allows us to establish certain necessary admissibility conditions on
the involved parameters. Hence, in contrast to general state spaces considered in Chapter 1, we can here precisely describe F and R by means of a
certain parameter set (see Proposition 2.3.3 below). Moreover, we relate the
particular form of F and R to the notion of quasi-monotonicity.
For the proof of the main results of this section we first provide a convergence result for Fourier-Laplace transforms.
Lemma 2.3.1. Let (νn )n∈N be a sequence of measures on V with
Z
e−hu,ξi νn (dξ) < ∞ and lim Ln (u) = L(u), u ∈ K ∗ ,
Ln (u) =
n→∞
V
pointwise, for some finite function L on K ∗ , continuous at u = 0. Then
νn converges weakly to some finite measure ν on V and the Fourier-Laplace
transform converges for u ∈ K̊ ∗ ∪ {0} and v ∈ V to the Fourier-Laplace
transform of ν, that is,
Z
Z
−hu+i v,ξi
e−hu+i v,ξi ν(dξ).
lim
e
νn (dξ) =
n→∞
V
V
In particular, ν(V ) = limn→∞ νn (V ) and
Z
L(u) =
e−hu,ξi ν(dξ),
V
for all u ∈ K̊ ∗ ∪ {0}.
Proof. Since νn (V ) = Ln (0) is bounded, we know that νn has a vague accumulation point ν, which is a finite measure on V .
Since Ln (u) < ∞ on K ∗ , it follows by well-known regularity properties
of Laplace transforms (see, e.g., Duffie et al. [2003, Lemma A.2]) that the
functions Ln admit an analytic extension to the strip K̊ ∗ + i V , still denoted
by Ln :
Z
(u + i v) 7→ Ln (u + i v) =
e−hu+i v,ξi νn (dξ).
V
Moreover, pointwise convergence of the finite convex functions Ln to L on
K ∗ implies that this convergence is in fact uniform on compact subsets of
2.3. Necessary Conditions
53
K̊ ∗ (see, e.g., Rockafellar [1997, Theorem 10.8]). Hence the functions Ln are
uniformly bounded on compact subsets of K̊ ∗ and since |Ln (u + i v)| ≤ Ln (u),
also on compact subsets of K̊ ∗ + i V . Moreover, since K ∗ is generating, K̊ ∗
is a set of uniqueness in K̊ ∗ + i V . It therefore follows from Vitali’s theorem
(see Narasimhan [1971, Chapter 1, Proposition 7]) that the analytic functions
Ln converge uniformly on compact subsets of K̊ ∗ + i V to an analytic limit
thereon. By Lévy’s continuity theorem, we therefore know that for any u ∈ K̊ ∗
the finite measures exp(−hu, ξi)νn (dξ) converge weakly to a limit, which by
uniqueness of the weak limit has to equal exp(−hu, ξi)ν(dξ). Whence the
only vague accumulation point of νn is ν. Vague convergence implies weak
convergence if mass is conserved. Continuity of L(u) at u = 0 implies this
mass conservation. Indeed, by weak convergence of e−hεu,ξi νn for u ∈ K̊ ∗ and
ε > 0, we arrive at
Z
Z
−hεu,ξi
e
νn (dξ) =
e−hεu,ξi ν(dξ)
L(εu) = lim
n→∞ V
VZ
Z
e−hεu,ξi 1{hu,ξi>0} ν(dξ).
e−hεu,ξi 1{hu,ξi≤0} ν(dξ) +
=
V
V
By dominated convergence we thus obtain for ε → 0
Z
L(0) =
ν(dξ),
V
which is the desired mass conservation. Hence we have weak convergence
of the measures νn , which means in turn convergence of the Fourier-Laplace
transform on K̊ ∗ ∪ {0} + i V .
2.3.1
Lévy Khintchine Form of F and R
We are now prepared to provide the particular functional form of F and R for
affine processes on cones. Due to this assumption on the state space, we are
now able to state more specific restrictions on the involved parameters. As in
Chapter 1, χ : V → V denotes some bounded continuous truncation function
with χ(ξ) = ξ in a neighborhood of 0.
Proposition 2.3.2. Let X be an affine process on K. Then the functions F
and R as defined in (2.8) are of the form
Z
F (u) = hb, ui + c −
e−hu,ξi − 1 m(dξ),
(2.11)
K
Z
1
R(u) = − Q(u, u) + B > (u) + γ −
e−hu,ξi − 1 + hχ(ξ), ui µ(dξ), (2.12)
2
K
where
54
Chapter 2. Affine Processes on Proper Convex Cones
(i) b ∈ K,
(ii) c ∈ R+ ,
(iii) m is a Borel measure on K satisfying m({0}) = 0 and
Z
(kξk ∧ 1) m(dξ) < ∞.
K
(iv) Q : V × V → V is a symmetric bilinear function with Q(v, v) ∈ K ∗ for
all v ∈ V ,
(v) B > : V → V is a linear map,
(vi) γ ∈ K ∗ ,
(vii) µ is a K ∗ -valued σ-finite Borel measure on K satisfying µ({0}) = 0 and
Z
kξk2 ∧ 1 hx, µ(dξ)i < ∞ for all x ∈ K.
K
Proof. In order to derive the particular form of F and R with the above
parameter restrictions, we follow the approach of Keller-Ressel [2009, Theorem
2.6]. Note that the t-derivative of Pt e−hu,xi at t = 0 exists for all x ∈ K and
u ∈ K̊ ∗ , since
lim
t↓0
Pt e−hu,xi − e−hu,xi
e−φ(t,u)−hψ(t,u),xi − e−hu,xi
= lim
t↓0
t
t
= (−F (u) − hR(u), xi)e−hu,xi
(2.13)
is well-defined by Theorem 2.2.5. Moreover, we can also write
Pt e−hu,xi − e−hu,xi
t↓0
te−hu,xi
Z
1
−hu,ξ−xi
e
pt (x, dξ) − 1
= lim
t↓0 t
K
Z
1
pt (x, K) − 1
−hu,ξi
= lim
e
− 1 pt (x, dξ + x) +
.
t↓0
t K−x
t
−F (u) − hR(u), xi = lim
By the above equalities and the fact that pt (x, K) ≤ 1, we then obtain for
u=0
0 ≥ lim
t↓0
pt (x, K) − 1
= −F (0) − hR(0), xi.
t
2.3. Necessary Conditions
55
Setting F (0) = c and R(0) = γ yields c ∈ R+ and γ ∈ K ∗ , hence (ii) and (vi).
We thus obtain
Z
1
−(F (u) − c) − hR(u) − γ, xi = lim
e−hu,ξi − 1 pt (x, dξ + x). (2.14)
t↓0 t K−x
For every fixed t > 0, the right hand side of (2.14) is the logarithm of the
Laplace transform of a compound Poisson distribution supported on K − R+ x
with intensity pt (x, K)/t and compounding distribution pt (x, dξ +x)/pt (x, K).
Concerning the support, note that the compounding distribution is concentrated on K − x, which implies that the compound Poisson distribution has
support on the convex cone K − R+ x. By Lemma 2.3.1, the pointwise convergence of (2.14) for t → 0 to some function being continuous at 0 implies weak
convergence of the compound Poisson distributions to some infinitely divisible
probability distribution ν(x, dy) supported on K − R+ x. Indeed, this follows
from the fact that any compound Poisson distribution is infinitely divisible and
the class of infinitely divisible distributions is closed under weak convergence
(see Sato [1999, Lemma 7.8]). Again, by Lemma 2.3.1, the Laplace transform
of ν(x, dy) is then given as exponential of the left hand side of (2.14).
In particular, for x = 0, ν(0, dy) is an infinitely divisible distribution with
support on the cone K. By the Lévy–Khintchine formula on proper cones
(see, e.g., Skorohod [1991, Theorem 3.21]), its Laplace transform is therefore
of the form
Z
−hu,ξi
exp −hb, ui + (e
− 1)m(dξ) ,
K
where b ∈ K and m is a Borel measure supported on K with m({0}) = 0 such
that
Z
(kξk ∧ 1) m(dξ) < ∞,
K
yielding (iii). Therefore,
Z
F (u) = hb, ui + c −
(e−hu,ξi − 1)m(dξ).
K
We next obtain the particular form of R. Observe that for each x ∈ K
and k ∈ N,
exp (−(F (u) − c)/k − hR(u) − γ, xi)
1
is the Laplace transform of the infinitely divisible distribution ν(kx, dy)∗ k ,
where ∗ k1 denotes the k1 convolution power. For k → ∞, these Laplace transforms obviously converge to exp(−hR(u) − γ, xi) pointwise in u. Using again
56
Chapter 2. Affine Processes on Proper Convex Cones
the same arguments as before (an application of Lemma 2.3.1 as below equa1
tion (2.14)), we can deduce that ν(kx, dy)∗ k converges weakly to some infinitely divisible distribution L(x, dy) on K − R+ x with Laplace transform
exp(−hR(u) − γ, xi) for u ∈ K ∗ .
By the Lévy-Khintchine formula on V (see Sato [1999, Theorem 8.1]), the
characteristic function of L(x, dy) has the form
b u) = exp
L(x,
Z
+
1
hu, A(x)ui + hB(x), ui
2
!
ehu,ξi − 1 − hχ (ξ) , ui M (x, dξ) ,
(2.15)
V
for u ∈ i V , where, for every x ∈ D, A(x) ∈ S+ (V ) is a symmetric positive
semidefinite linear operator on V , B(x) ∈ V , M (x, ·) a Borel measure on V
satisfying M (x, {0})
Z
(kξk2 ∧ 1)M (x, dξ) < ∞,
V
and χ some appropriate truncation function. Furthermore, by Sato [1999,
Theorem 8.7],
Z
Z
Z
1
t→0
f (ξ)m(dξ) +
f (ξ)M (x, dξ)
(2.16)
f (ξ) pt (x, dξ + x) −→
t
V
V
V
holds true for all f : V → R which are bounded, continuous and vanishing on
a neighborhood of 0. We thus conclude that M (x, dξ) has support in K − x.
b u) admits an analytic extension to
Therefore, the characteristic function L(x,
∗
K + i V , which then has to coincide with the Laplace transform for u ∈ K ∗ .
Hence, for all x ∈ K,
1
− hR(u) − γ, xi = hu, A(x)ui − hB(x), ui
Z2
+
e−hu,ξi − 1 + hχ (ξ) , ui M (x, dξ) ,
u ∈ K ∗ . (2.17)
V
As the left side of (2.17) is linear in the components of xRand as K is generating, it follows that x 7→ A(x), x 7→ B(x) as well as x 7→ E (kξk2 ∧ 1)M (x, dξ)
for every E ∈ B(V ) are restrictions of linear maps on V . In particular, Condition (v) follows immediately. Moreover, hu, A(x)vi can be written as
hu, A(x)vi = hx, Q(u, v)i,
(2.18)
2.3. Necessary Conditions
57
where Q : V ×V → V is a symmetric bilinear function satisfying Q(v, v) ∈ K ∗
for all v ∈ V , since A(x) is a positive semidefinite operator. This therefore
yields (iv). Similarly, we have for all E ∈ B(V )
Z
Z
2
(kξk ∧ 1)M (x, dξ) = (kξk2 ∧ 1)hx, µ(dξ)i,
E
E
where µ is a K ∗ -valued Borel measure on V , satisfying µ({0}) = 0 and
Z
kξk2 ∧ 1 hx, µ(dξ)i < ∞,
for all x ∈ K.
V
Hence it only remains to prove that supp(µ) ⊆ K. In (2.16) take now x = n1 y
for some y ∈ K with kyk = 1 and nonnegative functions f = fn ∈ Cb (V )
with fn = 0 on K − n1 y. Then, for each n, the left side of (2.16) is zero, since
pt ( n1 y, ·) is concentrated on K − n1 y. As supp(m) ⊆ K, the first integral on
the right vanishes as well. Hence
Z
Z
1
1
fn (ξ)M
fn (ξ)
0=
y, dξ =
y, µ(dξ)
n
n
V
V
for any nonnegative function fn ∈ Cb (V ) with fn = 0 on K − n1 y implies that
supp(µ) ⊆ K − n1 y for each n. Thus we can conclude that supp(µ) ⊆ K, which
proves (vii). Due to the definition of Q and µ together with (2.17), R(u) is
clearly of form (2.12).
2.3.2
Parameter Restrictions
In the following we continue the analysis of the function R and derive further
restrictions on the involved parameters Q, B > and µ.
Proposition 2.3.3. Let X be an affine process on K with R of form (2.12) for
some Q, B > , γ and µ satisfying the conditions of Proposition 2.3.2 (iv)-(vii).
Then, for any u ∈ K ∗ and x ∈ K with hu, xi = 0, we have
(i) hx, Q(u, v)i = 0 for all v ∈ V ,
(ii)
R
K
hχ(ξ), uihx, µ(dξ)i < ∞,
(iii) hx, B > (u)i −
R
K
hχ(ξ), uihx, µ(dξ)i ≥ 0.
58
Chapter 2. Affine Processes on Proper Convex Cones
Proof. Let u ∈ K ∗ and x ∈ K with hu, xi = 0 be fixed. Define the linear
map U : V → R, v 7→ hu, vi. As established in the proof of Proposition 2.3.2,
−hR(u) − γ, xi is the Laplace transform of an infinitely divisible distribution L(x, dy) supported on K − R+ x. Similar to (2.17), we denote the Lévy
triplet of L(x, dy) by (A(x), B(x), M (x, dξ)). Let now Yx be a random variable with distribution L(x, dy). Then the distribution of U (Yx ) = hu, Yx i,
which we denote by Lu (x, dy), is again infinitely divisible and supported on
R+ . From Sato [1999, Proposition 11.10] we then infer that the Lévy triplet
(au (x), bu (x), νu (x, dξ)) of Lu (x, dy) with respect to some truncation function
χ
e on R is given by
au (x) = hu, A(x)ui,
Z
bu (x) = hB(x), ui +
(e
χ(hu, ξi) − hχ(ξ), ui) U∗ M (x, dξ),
K
νu (x, dξ) = U∗ M (x, dξ).
By the
R Lévy Khintchine formula on R+ , we conclude that au (x) = 0, bu (x) ≥ 0
and K (kξk ∧ 1)U∗ M (x, dξ) < ∞. The last condition already implies (ii) and
allows to choose χ
e = 0. Moreover, bu (x) ≥ 0 yields (iii). From
E
Dp
p
A(x)u, A(x)u
0 = au (x) = hu, A(x)ui =
it follows that hv, A(x)ui = 0 for all v ∈ V . Hence relation (2.18) implies (i).
Remark 2.3.4. For an interpretation of the conditions of Proposition 2.3.3
we refer to Section 3.3.
2.3.3
Quasi-monotonicity
Quasi-monotonicity plays a crucial role in comparison theorems for ordinary
differential equations and thus appears naturally in the setting of affine processes. As we shall see in Section 2.4, it is needed to establish global existence and uniqueness for the ordinary differential equations defined in (2.9)
and (2.10). In the following we prove that the function R, as given in (2.12),
is quasi-monotone increasing if the conditions of Proposition 2.3.3 (i)-(iii) are
satisfied.
Definition 2.3.5 (Quasi-monotonicity). Let U be a subset of V . A function
f : U → V is called quasi-monotone increasing (with respect to K ∗ and the
induced order ) if, for all u, v ∈ U and x ∈ K satisfying u v and hu, xi =
hv, xi,
hf (u), xi ≤ hf (v), xi.
2.3. Necessary Conditions
59
Accordingly, we call f quasi-constant if both f and −f are quasi-monotone
increasing.
Remark 2.3.6. Note that in a one-dimensional vector space any function is
quasi-monotone. It is only in dimension greater than one that the notion of
quasi-monotonicity becomes meaningful.
Proposition 2.3.7. Let R be of form (2.12) for some Q, B > , γ and µ satisfying the conditions of Proposition 2.3.2 (iv)-(vii) and Proposition 2.3.3 (i)-(iii).
Then R is quasi-monotone increasing on K ∗ .
Proof. Let δ > 0, and define
Z
1
>
e−hu,ξi − 1 + hχ(ξ), ui µ(dξ)
R (u) = − Q(u, u) + B (u) + γ −
2
{kξk≥δ}∩K
Z
1
hχ(ξ), uiµ(dξ)
= − Q(u, u) + γ + B > (u) −
2
{kξk≥δ}∩K
Z
−
e−hu,ξi − 1 µ(dξ).
δ
{kξk≥δ}∩K
(2.19)
Take now some u, v ∈ K ∗ and x ∈ K such that u v and hu, xi = hv, xi.
Due to Condition (i) of Proposition 2.3.3, we then have hQ(v − u, w), xi = 0
for all w ∈ V . As Q is a bilinear function, this is equivalent to hQ(v, w), xi =
hQ(u, w), xi for all w ∈ V . Inserting w = u and w = v, we obtain by the
symmetry of Q
hQ(v, v), xi = hQ(u, u), xi,
whence the map u 7→ − 21 Q(u, u) + γ is quasi-constant. Condition (iii) of
Proposition 2.3.3 directly yields that
Z
>
u 7→ B (u) −
hχ(ξ), uiµ(dξ)
{kξk≥δ}∩K
is a quasi-monotone increasing linear map on K ∗ . Finally, the quasi-monotonicity of
Z
u 7→
1 − e−hu,ξi µ(dξ)
{kξk≥δ}∩K
is a consequence of the monotonicity of the exponential map and supp(µ) ⊆ K.
By dominated convergence, we have limδ→0 Rδ (u) = R(u) pointwise for each
u ∈ K ∗ . Hence the quasi-monotonicity carries over to R. Indeed, we have for
all δ > 0, hRδ (v) − Rδ (u), xi ≥ 0. Thus
hRδ (v) − Rδ (u), xi → hR(v) − R(u), xi ≥ 0
as δ → 0, which proves that R is quasi-monotone increasing.
60
2.4
Chapter 2. Affine Processes on Proper Convex Cones
The Generalized Riccati Equations
In order to prove the existence of affine processes for a given parameter set
which satisfies the conditions of Proposition 2.3.2 and Proposition 2.3.3, we
shall heavily rely on the following existence and uniqueness result for the generalized Riccati equations (2.9) and (2.10), where F and R are given by (2.11)
and (2.12). Indeed, in the case of general proper convex cones, this allows
us to prove existence of affine pure jumps processes (see Section 2.5). In
the particular case of affine processes on symmetric cones, which we study in
Chapter 3, we obtain, using Theorem 2.4.3 below, existence of affine processes
for any given parameter set (see Section 3.4.2).
For the analysis of the generalized Riccati equations (2.9) and (2.10) we
shall use the concept of quasi-monotonicity, as introduced above, several times.
Indeed, the proof of Theorem 2.4.3 below is based to a large extent on the
following comparison result for ordinary differential equations, which can be
deduced from a more general theorem proved by Volkmann [1973].
Theorem 2.4.1. Let U ⊂ V be an open set. Let f : [0, T ) × U → V be a
continuous locally Lipschitz map such that f (t, ·) is quasi-monotone increasing
on U for all t ∈ [0, T ). Let 0 < t0 ≤ T and g, h : [0, t0 ) → U be differentiable
maps such that g(0) h(0) and
∂t g(t) − f (t, g(t)) ∂t h(t) − f (t, h(t)),
0 ≤ t < t0 .
Then we have g(t) h(t) for all t ∈ [0, t0 ).
The following estimate is needed to establish the existence of a global
solution of (2.10).
Lemma 2.4.2. Let R be of form (2.12) for some Q, B > , γ and µ satisfying
the conditions of Proposition 2.3.2 (iv)-(vii). Then
R(u) B > (u) + γ + µ(K ∩ {kξk > 1}).
Proof. We may assume without loss of generality that the truncation function
χ takes the form χ(ξ) = 1{kξk≤1} ξ (otherwise we can adjust B > (u) accordingly). Then, for all u ∈ K ∗ , we have
Z
1
>
e−hu,ξi − 1 + hξ, ui µ(dξ)
R(u) = − Q(u, u) + B (u) + γ −
2
{z
}
K∩{kξk≤1} |
≥0
Z
−
e−hu,ξi − 1 µ(dξ)
K∩{kξk>1}
1
− Q(u, u) + B > (u) + γ + µ(K ∩ {kξk > 1})
2
>
B (u) + γ + µ(K ∩ {kξk > 1}),
2.4. The Generalized Riccati Equations
where we use −
R
K∩{kξk>1}
61
R
e−hu,ξi − 1 µ(dξ) K∩{kξk>1} µ(dξ).
Here is our main existence and uniqueness result for the generalized Riccati
differential equations (2.9)–(2.10).
Theorem 2.4.3. Let F and R be of form (2.11) and (2.12) such that the
conditions of Proposition 2.3.2 and 2.3.3 are satisfied. Then, for every u ∈
K̊ ∗ , there exists a unique global R+ × K̊ ∗ -valued solution (φ, ψ) of (2.9)–(2.10).
Moreover, φ(t, u) and ψ(t, u) are analytic in (t, u) ∈ R+ × K̊ ∗ .
Proof. We only have to show that, for every u ∈ K̊ ∗ , there exists a unique
global K̊ ∗ -valued solution ψ of (2.10), since φ is then uniquely determined by
integrating (2.9).
Let u ∈ K̊ ∗ . Since R is analytic on K̊ ∗ (see, e.g., Duffie et al. [2003,
Lemma A.2]), standard ODE results (see, e.g., Dieudonné [1969, Theorem
10.4.5]) yield that there exists a unique local K̊ ∗ -valued solution ψ(t, u) of
(2.10) for t ∈ [0, t∞ (u)), where
t∞ (u) = lim inf{t ≥ 0 | kψ(t, u)k ≥ k or ψ(t, u) ∈ ∂K ∗ } ≤ ∞.
k→∞
It thus remains to show that t∞ (u) = ∞. Analyticity of ψ(t, u) and φ(t, u) in
(t, u) ∈ R+ × K̊ ∗ then follows from Dieudonné [1969, Theorem 10.8.2].
Since R may not be Lipschitz continuous at ∂K ∗ (see Remark 2.4.4 below),
we first have to regularize it. We thus define
Z
1
>
e
e−hu,ξi − 1 + hξ, ui µ(dξ).
R(u) = − Q(u, u) + B (u) + γ −
2
K∩{kξk≤1}
e is analytic on V . Hence, for all u ∈ V , there exists a unique local
Then R
V -valued solution ψe of
e u)
∂ ψ(t,
e u)),
e ψ(t,
= R(
∂t
e u) = u,
ψ(0,
for all t ∈ [0, e
t∞ (u)) with maximal lifetime
e u)k ≥ k} ≤ ∞.
e
t∞ (u) = lim inf{t ≥ 0 | kψ(t,
k→∞
Consider now the normal cone of K ∗ at u ∈ ∂K ∗ , consisting of inward pointing
normal vectors, that is,
NK ∗ (u) = {x ∈ K | hu, xi = 0},
u 6= 0,
62
Chapter 2. Affine Processes on Proper Convex Cones
and NK ∗ (0) = K (see, e.g., Hiriart-Urruty and Lemaréchal [1993, Example
III.5.2.6], except for a change of the sign). The conditions of Proposition 2.3.3
thus imply that
e
hR(u),
xi ≥ 0,
e is clearly Lipschitz continuous, it follows from Walfor all x ∈ NK ∗ (u). Since R
e u) ∈ K ∗ for all t < e
ter [1993, Theorem III.10.XVI] that ψ(t,
t∞ (u) and u ∈ K ∗ .
Let us now define
∂y(t, u)
= B > (y(t, u)) + γ,
∂t
y(0, u) = u.
(2.20)
Then we have by Lemma 2.4.2 for t < e
t∞ (u)
e u)
∂ ψ(t,
e u)) = ∂y(t, u) − B > (y(t, u)) − γ ∂y(t, u) − R(y(t,
e ψ(t,
e
− R(
u)).
∂t
∂t
∂t
Volkmann’s comparison Theorem 2.4.1 thus implies for all x ∈ K
e u), xi ≤ hy(t, u), xi,
hψ(t,
t ∈ [0, e
t∞ (u)).
e u) lies in K ∗ up to its lifetime, the left hand side is nonnegative for all
As ψ(t,
x ∈ K. Moreover, the linear ODE (2.20) admits a global solution. This and
the fact that K is generating yields e
t∞ (u) = ∞ for all u ∈ K ∗ .
e is quasi-monotone increasing on K ∗ .
Moreover, by Proposition 2.3.7, R
Hence another application of Theorem 2.4.1 yields
e u) ψ(t,
e v),
0 ψ(t,
t ≥ 0,
for all 0 u v.
e u) is also analytic in u, Lemma 2.2.1 implies that
Therefore and since ψ(t,
e u) ∈ K̊ ∗ for all (t, u) ∈ R+ × K̊ ∗ .
ψ(t,
We now carry this over to ψ(t, u) and assume without loss of generality,
as in the proof of Lemma 2.4.2, that the truncation function χ takes the form
χ(ξ) = 1{kξk≤1} ξ. Then
Z
e
R(u) − R(u) = −
e−hu,ξi − 1 µ(dξ) 0, u ∈ K ∗ .
K∩{kξk>1}
Hence, for u ∈ K̊ ∗ and t < t∞ (u), we have
e u)
∂ ψ(t,
e u)) = ∂ψ(t, u) − R(ψ(t, u)) ∂ψ(t, u) − R(ψ(t,
e ψ(t,
e
− R(
u)).
∂t
∂t
∂t
Theorem 2.4.1 thus implies
e u) ∈ K̊ ∗ ,
ψ(t, u) ψ(t,
t ∈ [0, t∞ (u)).
2.5. Construction of Affine Pure Jump Processes
63
Hence t∞ (u) = limk→∞ inf{t ≥ 0 | kψ(t, u)k ≥ k}. Using again Lemma 2.4.2
e we conclude
and the comparison argument with a linear ODE similar as for ψ,
that t∞ (u) = ∞, as desired.
Remark 2.4.4. Note that quasi-monotonicity just means that R is “inward
pointing” close to the boundary K ∗ . If R was Lipschitz continuous on K ∗ , Walter [1993, Theorem III.10.XVI] would imply the invariance of K ∗ with respect
to (2.10) right away. The difficulty arises from the fact that the map R might
fail to be Lipschitz continuous at ∂K ∗ (see the one-dimensional counterexample in Duffie et al. [2003, Example 9.3]), even though it is analytic on the
interior K̊ ∗ . Here quasi-monotonicity plays the decisive role. It leads to the
phenomenon that ψ(t, u) stays away from the boundary ∂K ∗ for u ∈ K̊ ∗ , which
is of crucial importance in our analysis.
2.5
Construction of Affine Pure Jump Processes
For affine processes on generating convex proper cones without diffusion component, that is, Q = 0, the existence question can be handled entirely as in the
n−m
. By following
case of affine processes on the canonical state space Rm
+ ×R
the lines of Duffie et al. [2003, Section 7], we here prove existence of affine
pure jump processes for a given parameter set, which satisfies the conditions of
Proposition 2.3.2 and Proposition 2.3.3 with the additional assumption Q = 0.
We call a function f : K ∗ → R of Lévy-Khintchine form on K ∗ if
Z
f (u) = hb, ui − (e−hu,ξi − 1)m(dξ),
K
where b ∈ K and m is a Borel measure supported on K such that
Z
(kξk ∧ 1) m(dξ) < ∞.
K
Recall that a distribution on K is infinitely divisible if and only if its Laplace
transform takes the form e−f (u) , where f is of the above form. This means –
similarly as in the case of R+ – that Lévy processes on cones can only be of
finite variation.
As in Duffie et al. [2003], let us introduce the sets
C := {f + c | f : K ∗ → R is of Lévy-Khintchine form on K ∗ , c ∈ R+ },
CS := {ψ | u 7→ hψ(u), xi ∈ C for all x ∈ K }.
The following assertion can be obtained easily by mimicking the proofs
of Duffie et al. [2003, Proposition 7.2 and Lemma 7.5].
64
Chapter 2. Affine Processes on Proper Convex Cones
Lemma 2.5.1. We have,
(i) C, CS are convex cones in C(K ∗ ).
(ii) φ ∈ C, ψ ∈ CS imply φ(ψ) ∈ C.
(iii) ψ, ψ1 ∈ CS imply ψ1 (ψ) ∈ CS .
(iv) If φk ∈ C converges pointwise to a continuous function φ on K̊ ∗ , then
φ ∈ C and φ has a continuous extension to K ∗ . A similar statement
holds for sequences in CS .
(v) Let R be of form (2.12) and let Rδ be defined as in (2.19) such that the
involved parameters satisfy the conditions of Proposition 2.3.3. Then Rδ
converges to R locally uniformly as δ → 0.
Proposition 2.5.2. Let F and R be of form (2.11) and (2.12) such that
the involved parameters satisfy the conditions of Proposition 2.3.2 and 2.3.3.
Then, for all t ≥ 0, the solutions (φ(t, ·), ψ(t, ·)) of (2.9) and (2.10) lie in
(C, CS ).
Proof. Suppose first that
Z
(kξk ∧ 1) hx, µ(dξ)i < ∞,
for all x ∈ K.
(2.21)
Then equation (2.10) is equivalent to the integral equation
Z t
e>
e> t
B
e
u))ds,
eB (t−s) (R(ψ(s,
ψ(t, u) = e (u) +
(2.22)
K
0
e
e > (u) and B
e > ∈ L(V ) is given by
where R(u) = R(u)
+B
Z
>
>
e
B (u) := B (u) − hχ(ξ), uiµ(dξ).
K
e>
Here, eB t (u) is the notation for the semigroup induced by
e > (y(t, u)),
∂t y(t, u) = B
y(0, u) = u.
Hence the variation of constants formula yields (2.22). Due to Proposie > is a linear drift which is “inward pointing” at the boundary
tion 2.3.3 (iii), B
e>
of K ∗ . This in turn is equivalent to eB t being a positive semigroup, that is,
e>
e>
e
eB t maps K ∗ into K ∗ . Therefore eB t ∈ CS and since R(u)
is given by
Z
e
R(u)
= γ − (e−hu,ξi − 1)µ(dξ)
K
2.5. Construction of Affine Pure Jump Processes
65
e ∈ CS .
with µ satisfying (2.21), we also have R
By a classical fixed point argument, the solution ψ(t, u) is the pointwise
limit of the sequence (ψ (k) (t, u))k∈N , for (t, u) ∈ R+ × K̊ ∗ , obtained by Picard
iteration
ψ (0) (t, u) := u,
ψ
(k+1)
(t, u) := e
e> t
B
Z
(u) +
t
e > (t−s)
eB
e (k) (s, u))ds,
(R(ψ
0
and due to Lemma 2.5.1 (i) and (iii), ψ (k) (t, ·) lies in CS for all k ∈ N. In view
of Lemma 2.5.1 (iv), the limit ψ(t, ·) thus lies in CS as well and there exists
a unique continuous extension of ψ on R+ × K ∗ . Since F ∈ C, we have by
Lemma 2.5.1 (ii)
Z
t
F (ψ(s, ·))ds ∈ C.
φ(t, ·) =
0
By applying Lemma 2.5.1 (v), the general case is then reduced to the former,
since µ1{kξk≥δ} clearly satisfies (2.21).
We are now prepared to prove existence of affine processes on generating
convex proper cones under the additional assumption Q = 0:
Proposition 2.5.3. Suppose that the parameters (Q = 0, b, B, c, γ, m, µ) satisfy the conditions of Proposition 2.3.2 and Proposition 2.3.3. Then there exists a unique affine process on K such that (2.1) holds for all (t, u) ∈ R+ ×K ∗ ,
where φ(t, u) and ψ(t, u) are solutions of (2.9) and (2.10) with F and R given
by (2.11) and (2.12).
Proof. By Proposition 2.5.2, (φ(t, ·), ψ(t, ·)) lie in (C, CS ). Hence, for all t ∈ R+
and x ∈ K, there exists an infinitely divisible sub-stochastic measure on K
with Laplace-transform e−φ(t,u)−hψ(t,u),xi . Moreover, the Chapman-Kolmogorov
equation holds in view of the flow property of φ and ψ, which implies the
assertion.
Chapter 3
Affine Processes on Symmetric
Cones
We will now considerably strengthen our structural assumptions on the cone
state space K and assume that K is a symmetric cone. This assumption allows
us to refine the conditions found in Proposition 2.3.2 and Proposition 2.3.3
and enables us to prove existence of affine processes on these particular state
spaces. Concerning the refinement of the necessary conditions, we build on our
results obtained in the setting of positive semidefinite matrices (see Cuchiero
et al. [2011a]), while we use a different approach for the treatment of the
existence question.
The setting of symmetric cones, which was introduced in the field of affine
processes by Grasselli and Tebaldi [2008], covers many important examples,
such as the cone Rn+ , the cone of positive semi-definite
matrices, and the
P
Lorentz cone Λn , defined by Λn := {x ∈ Rn | x21 − ni=2 x2i > 0, x1 > 0}. It also
imposes an additional algebraic structure on V and leads to a multiplication
operation ◦ : V × V → V , which endows V with the structure of a so-called
Euclidean Jordan Algebra. The cone K is then exactly the cone of squares in
this algebra, that is, K = {x ◦ x : x ∈ V }.
3.1
Symmetric Cones and Euclidean Jordan
Algebras
We start by explaining the fundamental definitions from the standard reference on symmetric cones and Euclidean Jordan algebras, Faraut and Korányi
[1994]. In order to give some intuition, we always illustrate them by means
of the Euclidean Jordan algebra of r × r real symmetric matrices, which we
denote by Sr . Let us start with the definition of a symmetric cone.
67
68
Chapter 3. Affine Processes on Symmetric Cones
Definition 3.1.1 (Symmetric cone). A convex cone K in an Euclidean space
(V, h·, ·i) of dimension n is called symmetric if it is
(i) homogeneous, which means that the automorphism group G(K) = {g ∈
GL(V ) | gK = K} acts transitively, that is, for all x, y ∈ K̊ there exists
an invertible linear map g : V → V that leaves K invariant and maps x
to y,
(ii) self-dual, that is, K ∗ = K.
A symmetric cone K is said to be irreducible if there do not exist non-trivial
subspaces V1 , V2 and symmetric cones K1 ⊂ V1 , K2 ⊂ V2 such that V is the
direct sum of V1 and V2 and K = K1 + K2 .
Example 3.1.2. For illustrative purposes, let us consider the vector space
. A scalar product on this space is given by
Sr of dimension n = r(r+1)
2
hx, yi = tr(xy), where tr denotes the usual matrix trace. The corresponding
symmetric cone is the set of positive semidefinite matrices, which we denote
by Sr+ . Moreover, we write Sr++ for the open cone of positive definite matrices.
Clearly, Sr+ is self-dual with respect to hx, yi = tr(xy) and its automorphism
group is given by
G(Sr+ ) = G ∈ GL(Sr ) | Gx = gxg > , g ∈ GL(Rr ) .
Since every matrix x ∈ Sr++ can be written as x = zz > , where z is an invertible
r × r matrix, it follows that Sr+ is homogeneous.
Note that a symmetric cone is automatically generating and pointed. As
already mentioned, symmetric cones are directly related to Euclidean Jordan
algebras, defined as follows:
Definition 3.1.3 (Euclidean Jordan algebra). A real Euclidean space (V, h·, ·i)
with a bilinear product ◦ : V × V → V : (x, y) 7→ x ◦ y and identity element e
is called an Euclidean Jordan algebra if
(i) V is a Jordan algebra with product ◦, that is, for all x, y ∈ V
(a)
x ◦ y = y ◦ x,
(b) x2 ◦ (x ◦ y) = x ◦ (x2 ◦ y),
(ii) and the Jordan product is compatible with the scalar product in the sense
that
hx ◦ y, zi = hy, x ◦ zi.
An Euclidean Jordan algebra is said to be simple if it does not contain any
non-trivial ideal.
3.1. Symmetric Cones and Euclidean Jordan Algebras
69
Remark 3.1.4. Note that the Jordan product is commutative by (a), but
in general not associative. Thus (b) is a genuine axiom. We have used x2
to denote the Jordan product x ◦ x. This should not cause confusion, even
when we use the same notation to denote powers of scalars and matrices. By
induction it is seen that V is a power associative algebra, that is, xm ◦ xn =
xn ◦ xm = xm+n , for all m, n ≥ 1.
Example 3.1.5. By defining the following product on the vector space of r ×r
real symmetric matrices
1
x ◦ y = (xy + yx),
2
it is easily verified that Sr is a Jordan algebra. Here xy denotes the usual matrix multiplication. Note also that the scalar product satisfies hx, yi = tr(xy) =
tr(yx) = tr(x ◦ y).
For an element x ∈ V we introduce the left-product operator, denoted by
L and defined by
L(x)y = x ◦ y.
Moreover, P denotes the so-called quadratic representation of V , given by
P (x) = 2L(x)2 − L(x2 ).
(3.1)
For both operators we have L = L> and P = P > . In the case of Sr , the
quadratic representation is given by P (x)y = xyx.
The one-to-one correspondence between Euclidean Jordan algebras and
symmetric cones is established in Faraut and Korányi [1994, Theorem III.3.1,
III.4.4 and III.4.5] and can be rephrased as follows:
Theorem 3.1.6. Let K be a symmetric cone in V . Then there exists a Jordan
product ◦ on V such that (V, ◦) is an Euclidean Jordan algebra, and
K = {x2 : x ∈ V }.
The symmetric cone is irreducible if and only if the associated Euclidean Jordan algebra is simple.
Example 3.1.7. In the case of Sr+ , the above theorem can easily be verified,
since
Sr+ = {x ◦ x = x2 | x ∈ Sr }.
Note that the Jordan product x ◦ x = x2 equals the matrix product xx = x2 in
this case. This is positive semidefinite, since the eigenvalues of x2 are given
by the squared eigenvalues of x.
70
Chapter 3. Affine Processes on Symmetric Cones
For further definitions and results on Jordan algebras and symmetric cones,
which we shall use in the subsequent sections, we refer to Faraut and Korányi
[1994] and Appendix A. There we give an overview of the most important
results of the theory of Jordan algebras and illustrate them by means of realvalued symmetric matrices. Our terminology and notation follows the book
of Faraut and Korányi [1994].
3.2
Refinement of the Necessary Conditions
Throughout this section we suppose that X is an affine process on some irreducible symmetric cone K and V denotes the associated simple Euclidean
Jordan algebra of dimension n and rank r, equipped with the natural scalar
product
h·, ·i : V × V → R, hx, yi := tr(x ◦ y).
For the notion of the trace, denoted by tr, we refer to Appendix A.1.1. We shall
also use the Peirce invariant d corresponding to the dimension of Vij , i < j,
as defined in (A.7). In our case of a simple Euclidean Jordan algebra, we then
have n = r + d2 r(r − 1). For the precise definitions we refer to Appendix A.
Using the additional Euclidean Jordan algebra structure, we aim to improve the parameter conditions derived in Proposition 2.3.2 and Proposition 2.3.3 such that we finally obtain conditions which guarantee existence
of affine processes on symmetric cones. The focus lies in particular on the
bilinear form Q corresponding to the linear diffusion part, on the linear jump
coefficient µ and on the constant drift part b.
3.2.1
Representation of the Diffusion Part
The following proposition establishes a direct relation between the bilinear
form Q satisfying the condition of Proposition 2.3.3 and the quadratic representation of V . For its proof we use the Peirce and spectral decomposition of
an Euclidean Jordan algebra, as introduced in Appendix A.1.2.
Proposition 3.2.1. Let V be a simple Euclidean Jordan algebra of rank r
and let Q : V × V → V be a symmetric bilinear function with Q(v, v) ∈ K for
all v ∈ V . Then Condition (i) of Proposition 2.3.3, that is,
hx, Q(u, v)i = 0 for all v ∈ V and u, x ∈ K with hu, xi = 0,
is satisfied if and only if
Q(u, u) = 4P (u)α.
(3.2)
3.2. Refinement of the Necessary Conditions
71
Here, P (u) is the quadratic representation of the Jordan algebra V , defined
in (3.1) and α ∈ K is determined by 4α = Q(e, e).
Remark 3.2.2. (i) In the case of V = Sr , the above proposition means
that Condition (i) of Proposition 2.3.3 is equivalent to Q(u, u) = 4uαu,
where α ∈ Sr+ .
(ii) The above assertion has also been obtained in Grasselli and Tebaldi
[2008], but has been proved differently therein.
Proof. We first assume that (3.2) is satisfied. Let u ∈ V be fixed. Then there
exists a Jordan frame p1 , . . . , pr (see Appendix A.1.2 for
Prthe precise definition)
such that its spectral decomposition is given by u = i=1 λi pi .
As p1 + · · · + pr = e, we can write Q(e, e) as
X
Q(e, e) = Q(p1 , p1 ) + Q(p2 , p2 ) + · · · + Q(pr , pr ) +
2Q(pi , pj ).
(3.3)
i<j
We now show that (3.3) is precisely the Peirce decomposition of Q(e, e) with respect to the Jordan frame p1 , . . . , pr . More precisely, we show that Q(pj , pj ) ∈
Vjj and Q(pi , pj ) ∈ Vij for each i, j ∈ {1, . . . , r}.
Let i 6= j, then clearly hpi , pj i = 0. From (3.2) we deduce that
hpi , Q(pj , pj )i = 0.
(3.4)
But Q(pj , pj ) ∈ K such that Q(pj , pj ) ◦ pi = 0 by Lemma A.2.1. Keeping j
fixed, we can subtract these equalities from Q(pj , pj ) ◦ e = Q(pj , pj ), running
through all i 6= j, and we arrive at Q(pj , pj ) ◦ pj = Q(pj , pj ). This shows that
Q(pj , pj ) ∈ V (pj , 1) = Vjj for all j ∈ {1, . . . , r}.
Let now i, j, k be arbitrary in {1, . . . , r}, but all distinct. Using again (3.2),
we see that hpi , Q(pk + pj , pk + pj )i = 0, and from Lemma A.2.1 it follows that
Q(pk + pj , pk + pj ) ◦ pi = 0. Thus
Q(pk , pj )◦pi =
1
(Q(pk + pj , pk + pj ) ◦ pi − Q(pk , pk ) ◦ pi − Q(pj , pj ) ◦ pi ) = 0
2
for any distinct i, j, k ∈ {1, . . . , r}. Keeping now k and j fixed, we can subtract
the equalities Q(pk , pj ) ◦ pi = 0 from the equality Q(pk , pj ) ◦ e = Q(pk , pj ),
running through all i distinct from both j and k, and obtain Q(pk , pj ) ◦ (pk +
pj ) = Q(pk , pj ). For symmetry reasons we must have Q(pk , pj )◦pk = Q(pk , pj )◦
pj and we thus conclude that
1
Q(pk , pj ) ◦ pk = Q(pk , pj ) ◦ pj = Q(pk , pj ).
2
72
Chapter 3. Affine Processes on Symmetric Cones
Equivalently, Q(pk , pj ) ∈ V (pk , 1/2) ∩ V (pj , 1/2) = Vkj , and we have shown
that (3.3) is the Peirce decomposition of Q(e, e) with respect to the Jordan
frame p1 , . . . , pr .
Define 4α := Q(e, e). As the projection onto Vii is given by the quadratic
representation P (pi ) and the projection onto Vij by 4L(pi )L(pj ), we can write
Q(pj , pj ) = 4P (pj )α and 2Q(pi , pj ) = 16L(pi )L(pj )α. Therefore,
Q(u, u) = λ21 Q(p1 , p1 ) + · · · + λ2r Q(pr , pr ) +
X
2λi λj Q(pi , pj )
i<j
!
= 4 λ21 P (p1 )α + · · · + λ2r P (pr )α +
X
λi λj 4L(pi )L(pj )α
i<j
= 4P
r
X
!
λi pi
α
i=1
= 4P (u)α,
and we have shown the first implication.
Concerning the other direction, let Q be given by Q(u, u) = 4P (u)α for
some α ∈ K. Using polarization, we then get
Q(u, v) = 2 (P (u + v) − P (u) − P (v)) α.
Take now some x, u ∈ K such that hx, ui = 0. By Lemma A.2.1 (ii), we have
u ◦ x = 0 and consequently
hx, u2 i = hx ◦ u, ui = 0,
which in turn implies u2 ◦x = 0. The definition of the quadratic representation
thus yields
hP (u)α, xi = hα, P (u)xi = hα, 2u ◦ (u ◦ x) − u2 ◦ xi = 0.
Similarly, we have hP (u+v)α, xi = hP (v)α, xi, which proves the assertion.
3.2.2
Linear Jump Coefficient
In this section, we show that the linear jump coefficient µ satisfying Condition (vii) of Proposition 2.3.2 and Condition (ii) of Proposition 2.3.3 necessarily integrates (kξk ∧ 1) if r > 1 and d > 0. The proof is based on an
idea of Mayerhofer [2011], who showed the corresponding result for positive
semidefinite matrices.
3.2. Refinement of the Necessary Conditions
73
Proposition 3.2.3. Let V be a simple Euclidean Jordan algebra with rank
r > 1 and Peirce invariant d > 0. Suppose that µ is a K-valued σ-finite Borel
measure on K satisfying µ({0}) = 0 and
Z
(kξk2 ∧ 1)hx, µ(dξ)i < ∞, for all x ∈ K.
K
Then Condition (ii) of Proposition 2.3.3, that is,
Z
hχ(ξ), uihx, µ(dξ)i < ∞ for all u, x ∈ K with hu, xi = 0,
(3.5)
K
implies
Z
(kξk ∧ 1)hx, µ(dξ)i < ∞,
for all x ∈ K.
K
Remark 3.2.4. It follows from the above proposition that only in the case
of R+ and the two-dimensional Lorentz cone, jumps of infinite variation are
possible. In all other cases we could now set the truncation function χ to
be 0 and adjust the linear drift accordingly. However, in order to cover all
irreducible cones, we shall keep the truncation function in the sequel.
Proof. Let p1 , . . . , pr be a fixed Jordan frame of V . Corresponding to the
Peirce decomposition (A.7), we can write for every z ∈ V
z=
r
X
zi pi +
i=1
X
zij ,
i<j
where zi ∈ R and zij ∈ Vij . For the K-valued measure µ, this means that,
for every i ∈ {1, . . . , r}, µi denotes a positive measure and, for i 6= j, µij is a
Vij -valued measure. Consider now, for some i 6= j, elements of the form
x = pi + pj + w,
u = pi + pj − w,
with w ∈ Vij such that kwk2 = 2, which implies that x, u ∈ ∂K and hu, xi = 0.
Assume without loss of generality that χ(ξ) = 1{kξk≤1} ξ. Since for every y ∈ K,
hy, µ(·)i is a positive measure supported on K, we have by (3.5)
Z
0≤
hξ, uihx, µ(dξ)i < ∞,
{kξk≤1}
Z
0≤
hξ, xihu, µ(dξ)i < ∞.
{kξk≤1}
74
Chapter 3. Affine Processes on Symmetric Cones
Thus there exists a positive constant C such that for all δ > 0
Z
hξ, uihx, µ(dξ)i < C,
0≤
{δ≤kξk≤1}
Z
hξ, xihu, µ(dξ)i < C.
0≤
(3.6)
(3.7)
{δ≤kξk≤1}
Summing up (3.6) and (3.7) and using the orthogonality of the Peirce decomposition, then yields
1
ξi µi (dξ) − hξij , wihw, µij (dξ)i + ξj µj (dξ)i
0≤
2
{δ≤kξk≤1}
Z
1
ξi µj (dξ) − hξij , wihw, µij (dξ)i + ξj µi (dξ) < 2C. (3.8)
+
2
{δ≤kξk≤1}
Z
Since µ is a K-valued measure on K, we have by Faraut and Korányi [1994,
Exercise IV.7 (b)] and the assumption kwk2 = 2
p
1
hξij , wihw, µij i ≤ kξij kkµij k ≤ 2 ξi ξj µi µj ,
2
which implies that both integrals in (3.8) are nonnegative. We can therefore
conclude that both of them are finite:
Z
1
0≤
ξi µi (dξ) − hξij , wihw, µij (dξ)i + ξj µj (dξ)i < 2C, (3.9)
2
{δ≤kξk≤1}
Z
1
0≤
ξi µj (dξ) − hξij , wihw, µij (dξ)i + ξj µi (dξ) < 2C. (3.10)
2
{δ≤kξk≤1}
Moreover, as hpi , pj i = 0 for i 6= j, we have as a direct consequence of (3.5)
Z
Z
0≤
hξ, pi ihpj , µ(dξ)i =
ξi µj (dξ)i < ∞, i 6= j.
(3.11)
{kξk≤1}
{kξk≤1}
As above, there thus exists a positive constant C1 such that for all δ > 0
Z
ξi µj (dξ) < C1 , i 6= j.
(3.12)
{δ≤kξk≤1}
Subtracting (3.10) from (3.12), then yields for all δ > 0
Z
1
−2C <
hξij , wihw, µij (dξ)i < 2C1 .
{δ≤kξk≤1} 2
3.2. Refinement of the Necessary Conditions
75
By (3.9), we therefore have for all δ > 0
Z
0≤
(ξi µi (dξ) + ξj µj (dξ)) < 2(C + C1 ).
{δ≤kξk≤1}
Together with (3.11), this implies for all i ∈ {1, . . . , r}
Z
0≤
ξi µi (dξ) < ∞.
{kξk≤1}
This then yields
Z
(kξk ∧ 1)hx, µ(dξ)i < ∞,
for all x ∈ K,
K
and proves the assertion.
3.2.3
The Special Role of the Constant Drift Part
This section is devoted to show that the constant drift term b, as defined in
Proposition 2.3.2 (i), of any affine process X on an irreducible symmetric cone
K necessarily satisfies
b d(r − 1)α,
where α is defined in Proposition 3.2.1. Recall that d denotes the Peirce
invariant and r the rank of V .
Before we actually prove this result, let us introduce some notation. We
shall consider the tensor product V ⊗ V ∗ , which we identify via the canonical
isomorphism with the vector space of linear maps on V denoted by L(V ).
Moreover, for an element A ∈ L(V ), we denote its trace by Tr(A).1 Observe
that Tr(A(u ⊗ u)) = hu, Aui. Indeed, by choosing a basis {eβ } of V , we have
X
Tr(A(u ⊗ u)) =
hA> eβ , (u ⊗ u)eβ i
β
X
=
hA> eβ , hu, eβ iui
β
X
=
hu, eβ ihA> eβ , ui
β
= hu, Aui.
1
In order to distinguish between elements of V and linear maps on V , we use the notations Tr(A) and Det(A) for A ∈ L(V ) and tr(x) and det(x) for elements in V (compare
Remark A.1.1).
76
Chapter 3. Affine Processes on Symmetric Cones
Let now A : K → S+ (V ) ⊂ L(V ) be the linear part of the diffusion characteristic, as introduced in (2.15). Recall that the symmetric bilinear function
Q was defined via (2.18), that is,
Tr(A(x)(u ⊗ u)) = hu, A(x)ui = hx, Q(u, u)i.
As shown in Proposition 3.2.1, we have Q(u, u) = 4P (u)α for some α ∈ K.
Hence
Tr(A(x)(u ⊗ u)) = hu, A(x)ui = 4hx, P (u)αi.
(3.13)
We can now define a second order differential operator for this expression
and introduce the following integro-differential operator for (complex-valued)
Cb2 (K)-functions.
1
∂
∂
Af (x) = Tr A(x)
⊗
f |x + hb + B(x), ∇f (x)i
2
∂x ∂x
Z
(3.14)
− (c + hγ, xi)f (x) +
(f (x + ξ) − f (x)) m(dξ)
K
Z
(f (x + ξ) − f (x) − hχ(ξ), ∇f (x)i) hx, µ(dξ)i,
+
K
where A(x) satisfies (3.13). The other parameters are specified in Proposition 2.3.2 and are supposed to satisfy the conditions of Proposition 2.3.3 and
Propsition 3.2.3. Note that for the family of functions {e−hu,xi | u ∈ K} this
expression corresponds to the pointwise t-derivative of Pt e−hu,xi at t = 0. This
is simply a consequence of the form of F and R, since
Pt e−hu,xi − e−hu,xi
lim
= (−F (u) − hR(u), xi)e−hu,xi = Ae−hu,xi
t↓0
t
for every x ∈ K. In view of Definition 1.4.1, A corresponds to the extended
generator of the affine process X.
The following lemma is proved by means of the Lévy–Khintchine formula
on R+ and is related to the positive maximum principle for the operator A.
Lemma 3.2.5. Let X be an affine process on K with constant drift parameter
b and linear diffusion part Q, as defined in Proposition 2.3.2 (i) and (iv).
Moreover, suppose that Q satisfies Q(u, u) = 4P (u)α for some α ∈ K. Then,
for any y ∈ ∂K, we have
∂
α det |y
hb, ∇ det(y)i + 2 y, P
∂x
(3.15)
1
∂
∂
= hb, ∇ det(y)i + Tr A(y)
⊗
det |y ≥ 0.
2
∂x ∂x
3.2. Refinement of the Necessary Conditions
77
Here, det(y) denotes the determinant of an element y ∈ V , as defined in
Appendix A.1.1, and A(x) is the linear part of the diffusion characteristic,
which satisfies
Tr(A(x)(u ⊗ u)) = hu, A(x)ui = hx, Q(u, u)i = 4hx, P (u)αi
(3.16)
for all u, x ∈ K.
Proof. Let y ∈ ∂K and let f ∈ Cc∞ (V ) be a function with f ≥ 0 on K and
f (x) = det(x) for all x in a neighborhood of y. Then, for any v ∈ R+ , the
function x 7→ (e−vf (x) − 1) lies in Cc∞ (V ), hence in particular in Sn , the space
of rapidly decreasing C ∞ -functions on V . As the Fourier transform is a linear
isomorphism on Sn , we can write
Z
−vf (y)
e
−1=
eihq,yi g(q)dq
V
for some g ∈ Sn . As a consequence of Theorem 1.5.4, Remark 2.1.2 and
Remark 2.2.4, we obtain by dominated convergence
Z
Pt (e−vf (y) − 1)
−vf (y)
lim
∂t |t=0 Pt eihq,yi g(q)dq
= ∂t |t=0 Pt (e
− 1) =
t↓0
t
V
Z
=
(F(i q) + hR(i q), yi) eihq,yi g(q)dq
ZV
(−F (− i q) − hR(− i q), yi) eihq,yi g(q)dq
=
ZV
=
Aeihq,yi g(q)dq = A(e−vf (y) − 1),
V
where A is defined in (3.14) and thus satisfies (−F (− i q)−hR(− i q), xi)eihq,xi =
Aeihq,xi . Hence the limit
Z
1
−vf (y)
A(e
− 1) = lim
(e−vf (ξ) − 1)pt (y, dξ)
t↓0 t K
Z
(3.17)
1
= lim
(e−vz − 1)pft (y, dz),
t↓0 t R
+
exists for any v ∈ R+ , where pft (y, dz) = f∗ pt (y, dz) is the pushforward of
pt (y, ·) under f , which is a probability measure supported on R+ . Using the
same arguments as in the proof of Proposition 2.3.2, we see that, for every fixed
t > 0, the right hand side of (3.17) is the logarithm of the Laplace transform
of a compound Poisson distribution supported on R+ with intensity 1/t and
compounding distribution pft (y, dz). The pointwise convergence of (3.17) for
78
Chapter 3. Affine Processes on Symmetric Cones
t → 0 to some function being continuous at 0 implies weak convergence of
the compound Poisson distributions to some infinitely divisible probability
distribution supported on R+ . Its Laplace transform is then given as the
exponential of the left hand side of (3.17).
Using now f (y) = 0 and the form of A given by (3.14), we have
v 7→ A(e−vf (y) − 1)
∂
α f |y
= 2v hy, P (∇f (y))αi − 2v y, P
∂x
Z
e−vf (y+ξ) − 1 m(dξ)
− vhb + B(y), ∇f (y)i +
K
Z
+
e−vf (y+ξ) − 1 + vhχ(ξ), ∇f (y)i hy, µ(dξ)i.
2
(3.18)
K
Note now that h∇ det(y), yi = 0 such that the admissibility Conditions (ii)
and (iii) of Proposition 2.3.3 imply2
Z
hχ(ξ), ∇ det(y)ihy, µ(dξ)i < ∞
K
and
e
B(y)
= hy, B > (∇ det(y))i −
Z
hχ(ξ), ∇ det(y)ihy, µ(dξ)i ≥ 0.
K
By the Lévy–Khintchine formula on R+ , hy, P (∇f (y))αi has to vanish, which
is the case due to Proposition 3.2.1 and the fact that h∇ det(y), yi = 0. Moreover, the coefficient of v in (3.18) has to be non-positive, that is,
Z
∂
2 y, P
α f |y + hb + B(y), ∇f (y)i − hχ(ξ), ∇f (y)ihy, µ(dξ)i ≥ 0.
∂x
K
e
Observing that y 7→ B(y)
is a polynomial of degree r, being positive for every
y ∈ ∂K, and that the polynomial
∂
y 7→ hb, ∇ det(y)i + 2 y, P
α det |y
∂x
is of degree r − 1, we obtain Condition (3.15).
2
R
By Proposition 3.2.3, we have K kχ(ξ)khy, µ(dξ)i < ∞ if r > 1 and d > 0. This means
that the above argument using h∇ det(y), yi = 0 is only relevant in the two-dimensional
Lorentz cone. Observe that for K = R+ , y = 0 anyway.
3.2. Refinement of the Necessary Conditions
79
Proposition 3.2.6. Let X be an affine process on K with constant drift parameter b ∈ K and diffusion parameter α ∈ K, which defines Q(u, u) through
Q(u, u) = 4P (u)α. Then
b d(r − 1)α,
where d denotes the Peirce invariant and r the rank of V .
Remark 3.2.7. Since d = 1 for V = Sr , the above conditions reads in this
case as
b (r − 1)α,
where α is specified in Remark 3.2.2 (i).
Proof. From Lemma 3.2.5 we have the necessary condition
1
∂
∂
hb, ∇ det(y)i + Tr A(y)
⊗
det |y ≥ 0
2
∂x ∂x
for any y ∈ ∂K. For x ∈ K̊ we can calculate the left hand side. Since
∇ det(x) = det(x)x−1 and dtd (x−1 +tu)|t=0 = −P (x−1 )u (Proposition A.2.2 (v)
and (iii)), we have
∂
∂
1
⊗
hb, ∇ det(x)i + Tr A(x)
det |x
2
∂x ∂x
−1 1
1
−1
−1
−1
= det(x) x , b + Tr A(x) x ⊗ x
− Tr A(x)P x
.
2
2
Using (3.16), Proposition A.2.2 (i) and Lemma 3.2.8 below, we thus obtain
−1 1
−1
−1
α − Tr A(x)P x
det(x) x , b + 2 x, P x
2
n
= det(x) x−1 , b + 2 x−1 , α − 2 x−1 , α
−1 −1 r = det(x) x , b − d(r − 1) x , α
= det(x) x−1 , b − d(r − 1)α .
As det(y)y −1 is also well-defined on ∂K, Condition (3.15) implies
b d(r − 1)α.
The following lemma is needed in the proof of the above proposition and
allows us to express Tr (A(x)P (x−1 )) in terms of α.
80
Chapter 3. Affine Processes on Symmetric Cones
Lemma 3.2.8. Let V be a simple Euclidean Jordan algebra of rank r and
with scalar product hx, yi = tr(x ◦ y) and let A(x) be defined by (3.16). Then
Tr A(x)P x−1
=4
n −1 x ,α
r
(3.19)
for any invertible x ∈ V .
Proof. Let p1 , . . . , pr be a Jordan frame ofP
V . Then the spectral decomposition
of an arbitrary element x is given by x = ri=1 λi pi , and P (x−1 ) can be written
as
P x
−1
=
r
X
λ−2
i P (pi ) +
i=1
X
−1
4λ−1
i λj L(pi )L(pj ).
(3.20)
i<j
Let now {eβ } be an orthonormal basis of V , where the basis elements are
chosen to lie in the subspaces corresponding to the Peirce decomposition, as
described in Section A.1.2. More precisely, for each i ≤ r, we choose one
basis element in Vii , which is in fact pi , and for each i < j, we choose d basis
elements in Vij , since the dimension of Vij is d.
By the definition of the trace Tr, we have
X
A(x)eβ , P (x−1 )eβ .
Tr A(x)P (x−1 ) =
β
In order to evaluate P (x−1 )eβ , we shall use
Vii = {x ∈ V | L(pk )x = δik x},
1
Vij = x ∈ V | L(pk )x = (δik + δjk )x ,
2
as derived in the proof of Faraut and Korányi [1994, Theorem IV.2.1]. This
implies for eβ ∈ Vij , i ≤ j,
1
L(pk )eβ = (δik eβ + δjk eβ ),
2
and hence for k < l,
1
e , if eβ ∈ Vkl ,
4 β
L(pl )L(pk )eβ =
0, otherwise,
(3.21)
P (pk )eβ =
eβ , if eβ ∈ Vkk ,
0, otherwise.
Note that this is obvious, since P (pk ) and 4L(pl )L(pk ) are the orthogonal
projections on Vkk and Vkl respectively (see Section A.1.2).
3.2. Refinement of the Necessary Conditions
81
Let now eβ ∈ Vij , i ≤ j be fixed. Then, using (3.20), the linearity of A
and (3.16), we obtain
* r
+
X
−1
A(x)eβ , P x−1 eβ =
λk A(pk )eβ , λ−1
i λj eβ
k=1
=
r
X
−1
λk λ−1
i λj 4hP (eβ )pk , αi
k=1
=2
−1
λ−1
.
i pi , α + λj pj , α
(3.22)
Here, the last equality follows from
1
(3.23)
P (eβ )pk = (δik pj + δjk pi ),
2
for eβ ∈ Vij , i ≤ j. For eβ ∈ Vii , (3.23) is simply a consequence of (3.21) and
for eβ ∈ Vij , i < j, we have by Faraut and Korányi [1994, Proposition IV.1.4
(i)]
1
P (eβ )pk = e2β ◦ (δik e + δjk e − pk ) = (pi + pj ) ◦ (δik e + δjk e − pk ),
2
which then leads to (3.23). By summing over all basis elements, we deduce
from (3.22)
r n
X
d
−1
Tr A(x)P x
4hx−1 , αi,
=
1 + (r − 1) 4 λ−1
i pi , α =
2
r
i=1
where the last equality follows from the fact that n = r + d2 r(r − 1).
Remark 3.2.9. If V is an associative Euclidean Jordan algebra, then (3.19)
can be derived in a simpler way. Defining a Jordan product by
1
x ◦ y = (xy + yx),
2
implies P (x)y = xyx and P (x, y)z = 12 (xzy + yzx) (see Faraut and Korányi
[1994, page 32]). Using now (3.16), yields
X
Tr A(x)P (x−1 ) =
A(x)eβ , P (x−1 )eβ
β
X =
4 x, P P (x−1 )eβ , eβ α
β
X =
2 x, x−1 eβ x−1 αeβ + eβ αx−1 eβ x−1
β
X n =
4 e2β , x−1 ◦ α = 4 x−1 , α ,
r
β
82
Chapter 3. Affine Processes on Symmetric Cones
P
since β e2β = nr e (see Faraut and Korányi [1994, e.g., Proof of Proposition
VI.4.1]).
In general, Lemma 3.2.8 cannot be proved by these simpler arguments,
since the Lorentz cone and the 3 × 3 Hermitian matrices over the octonions
are examples of non-associative simple Euclidean Jordan algebras.
3.3
Discussion of the Admissibility Conditions
In the following definition we summarize the above derived necessary parameter restrictions for affine processes on symmetric cones.
Definition 3.3.1 (Admissible parameters). An admissible parameter set
(α, b, B, c, γ, m, µ) (associated with a truncation function χ) for an affine process on an irreducible symmetric cone K consists of
• a linear diffusion coefficient
α ∈ K,
(3.24)
b d(r − 1)α,
(3.25)
c ∈ R+ ,
(3.26)
γ ∈ K,
(3.27)
• a constant drift term
• a constant killing rate term
• a linear killing rate coefficient
• a constant jump term: a Borel measure m on K satisfying
Z
m({0}) = 0 and
(kξk ∧ 1) m(dξ) < ∞,
(3.28)
K
• a linear jump coefficient: a K-valued σ-finite Borel measure µ on K
with µ({0}) = 0 such that the kernel
M (x, dξ) = hx, µ(dξ)i
(3.29)
satisfies
Z
(kξk2 ∧ 1)M (x, dξ) < ∞,
for all x ∈ K,
K
and
Z
hχ(ξ), uiM (x, dξ) < ∞ for all x, u ∈ K with hx, ui = 0,
K
(3.30)
3.3. Discussion of the Admissibility Conditions
83
• a linear drift coefficient: a linear map B > : V → V such that
Z
>
hx, B (u)i− hχ(ξ), ui M (x, dξ) ≥ 0 for all x, u ∈ K with hx, ui = 0.
K
(3.31)
Remark 3.3.2. Note that (3.30) and (3.31) can be considerably simplified if
dim V > 2. In this case r > 1 and d > 0 such that Proposition 3.2.3 allows to
set χ = 0.
Using the above definition, let us collect the results that we derived so far.
Theorem 3.3.3. Let X be an affine process on an irreducible symmetric cone
K. Then X is regular and has the Feller property. Moreover, φ and ψ satisfy
the generalized Riccati equations (2.9) and (2.10) and there exists an admissible parameter set (α, b, B, c, γ, m, µ) such that the functions F and R are of
the following form
Z
>
R(u) = −2P (u)α + B (u) + γ −
e−hξ,ui − 1 + hχ(ξ), ui µ(dξ), (3.32)
K
Z
F (u) = hb, ui + c −
e−hξ,ui − 1 m(dξ).
(3.33)
K
Proof. The result is a consequence of Theorem 2.2.2, Theorem 2.2.5, Proposition 2.3.2, Proposition 2.3.3, Proposition 3.2.1 and Proposition 3.2.6.
In order to give some intuition on the above defined admissibility conditions, we discuss and highlight some properties of the admissible parameter
set (α, b, B, c, γ, m, µ). In particular, we shall compare them with the welln−m
known admissibility conditions for the canonical state space Rm
and
+ ×R
m
the cone R+ (see Duffie et al. [2003, Definition 2.6]). Note that the latter is
a reducible symmetric cone, whose associated Euclidean Jordan algebra Rm
is of rank 1. We also exemplify the admissibility conditions by means of the
cone of r × r positive semidefinite matrices.
3.3.1
Diffusion
As we have already seen in Proposition 2.3.2, the diffusion term does not
admit a constant part if the state space is a general proper convex cone. This
is one difference to the mixed state space Rn × Rm
+ , where a constant diffusion
part is possible.
Condition (i) of Proposition 2.3.3, that is,
hx, Q(u, v)i = 0 for all v ∈ V and u, x ∈ K with hu, xi = 0,
84
Chapter 3. Affine Processes on Symmetric Cones
is simply a consequence of parallel diffusion behavior along the boundary of
the state space. In the case of symmetric cones, this then translates to
hu, A(x)ui = hx, Q(u, u)i = 4hx, P (u)αi,
α ∈ K,
u ∈ V.
(3.34)
This property is also in line with the admissibility conditions on the reducible symmetric
cone Rm
+ . In this case the diffusion part A(x) is of the form
Pm
A(x) = i=1 4αi ei xi . Here, αi ∈ R+ and ei denotes the matrix, where the
(ii)th entry is 1 and all others are 0. As the quadratic representation of Rm is
given by
P (x)y = (x21 y1 , . . . , x2m ym )> , x, y ∈ Rm ,
relation (3.34) also holds for the cone Rm
+ . In the case of positive semidefinite
matrices, the above simplifies to hu, A(x)ui = 4hx, uαui.
3.3.2
Drift
The remarkable drift Condition (3.25) can be explained by the fact that the
boundary of a symmetric cone is curved and kinked, implying this trade-off
between diffusion coefficient α and b. Indeed, this condition is a consequence
of the positive maximum principle for the (extended) generator A, defined
in (3.14) (see Lemma 3.2.5).
Note that, for the rank 1 Jordan algebra R and the symmetric cone R+ ,
the drift condition simply reduces to the non-negativity of b. Therefore we
+
only have b ∈ Rm
+ if K = Rm .
Let us now consider the cone of positive semidefinite r × r matrices with
r > 1. In this case the Peirce invariant equals 1, whence b (r − 1)α. In the
literature, the linear drift part B(x) is usually assumed to be of the form
B(x) = Hx + xH >
(3.35)
for some r × r matrix H. Then we have
hB(x), ui = hHx + xH > , ui = 0 for all x, u ∈ Sr+ with hx, ui = 0,
(3.36)
such that (3.31) is satisfied. Note that, due to Proposition 3.2.3, the truncation
function χ can set to be 0. Another possible specifcation of B(x), such that
Condition (3.31) still holds true, is
B(x) = Hx + xH > + Γ(x),
(3.37)
where Γ : Sr → Sr is a linear map satisfying Γ(Sr+ ) ⊆ Sr+ . Here is a simple
example where B(x) is of the form (3.37) but not of the usual form (3.35): let
r = 2 and
x22 x12
B(x) =
.
x12 x11
3.3. Discussion of the Admissibility Conditions
85
It can be easily checked that (3.31) is satisfied, while B(x) cannot be brought
into the form (3.35).
3.3.3
Killing
A necessary condition for an affine process on any convex proper cone to
be conservative is c = 0 and γ = 0. In the case of symmetric cones, it
can be proved as in Mayerhofer, Muhle-Karbe, and Smirnov [2011] that X is
conservative if and only if c = 0 and ψ(t, 0) ≡ 0 is the only K-valued local
solution of (2.10) for u = 0. The latter condition clearly requires that γ = 0.
A sufficient condition for X to be conservative is c = 0, γ = 0 and
Z
kξkM (x, dξ) < ∞,
for all x ∈ K.
K∩{kξk≥1}
Indeed, it can be shown similarly as in Duffie et al. [2003, Section 9] that the
latter property implies Lipschitz continuity of R(u) on K.
3.3.4
Jumps
For general convex proper cones, Condition (3.28) means that jumps described
by m, which can for instance appear at x = 0, should be of finite variation
entering the cone K, since infinite variation transversal to the boundary could
make the process leave the state space. Similarly, Condition (3.30) asserts
finite variation for the inward pointing direction. However, due to the geometry of irreducible symmetric cones in dimensions n > 2, such a behavior is no
longer possible and all jumps are in fact of finite total variation (see Proposition 3.2.3). Note however that in the case of R+ and the two-dimensional
Lorentz cone, the linear jump part can have infinite total variation (see [Duffie
et al., 2003, Equation (2.11)]).
Let us also remark that for r > 1 and d > 0, affine diffusion processes
cannot be approximated (in law) by pure jump processes, since this would
yield a contradiction to Condition (3.25). However, such an approximation is
possible for the canonical state space, since the rank of the Euclidean Jordan
algebra Rn is 1. This is explicitly exploited in the existence proof for affine
n−m
processes on Rm
(see Duffie et al. [2003, Section 7]).
+ ×R
86
3.4
Chapter 3. Affine Processes on Symmetric Cones
Construction of Affine Processes on Symmetric Cones
Throughout this section we use the same setting as in the previous one, that
is, we suppose that V is a simple Euclidean Jordan algebra of dimension n
and rank r and K is the associated irreducible symmetric cone. As before,
we assume that the scalar product on V is defined by hx, yi = tr(x ◦ y) and
the Peirce invariant d corresponds to the dimension of Vij , i < j, as defined in
Appendix (A.7). Again we refer to Faraut and Korányi [1994] and Appendix A
for results on Euclidean Jordan algebras.
3.4.1
Construction of Affine Diffusion Processes
The aim of this section is to establish existence of affine diffusion processes for
the following admissible parameter set (α, δα, 0, 0, 0, 0, 0) with δ ≥ d(r − 1).
To this end we consider the Riccati equations for φ and ψ associated to these
parameters and show that, for every (t, x) ∈ R+ × K, e−φ(t,u)−hψ(t,u),xi is the
Laplace transform of a probability distribution supported on K. It will turn
out that this probability distribution corresponds to the non-central Wishart
distribution in the case of positive semidefinite matrices. The existence of
such affine diffusion processes then follows from the semiflow property of φ
and ψ, which yields the Kolmogorov-Chapman equation for the transition
probabilities and thus the Markov property.
We start by establishing explicit solutions for the Riccati equations associated to the parameter set (α, δα, 0, 0, 0, 0, 0) with δ ∈ R+ .
Lemma 3.4.1. Let α ∈ K and δ ∈ R+ . Consider the following system of
Riccati differential equations for u ∈ K̊ and t ∈ R+
∂ψ(t, u)
= −2P (ψ(t, u))α,
∂t
∂φ(t, u)
= hδα, ψ(t, u)i,
∂t
ψ(0, u) = u ∈ K̊,
(3.38)
φ(0, u) = 0.
(3.39)
Then the solution is given by
ψ(t, u) = (u−1 + 2tα)−1 ,
√
δ
φ(t, u) = ln det e + 2tP ( α)u .
2
(3.40)
(3.41)
Moreover, u 7→ ψ(t, u) and u 7→ φ(t, u) are continuous at u = 0 for all t ∈ R+
with φ(t, 0) = 0 and ψ(t, 0) = 0.
3.4. Construction of Affine Processes on Symmetric Cones
87
Proof. Using
d
(x + tv)−1 = −P ((x + tv)−1 )v,
dt
which follows from Proposition A.2.2 (iii), one easily verifies that ψ(t, u) given
by (3.40) satisfies (3.38). Concerning φ(t, u), let us first show that
√
det(u) det(u−1 + 2tα) = det e + 2tP ( α)u .
(3.42)
Indeed, we have by Proposition A.2.2 (i), (iv) and Proposition A.1.3
√
√
det(u) det(u−1 + 2tα) = det(u) det P ( u−1 ) e + 2tP ( u)α
√
= det(u) det(u−1 ) det e + 2tP ( u)α
√
= det(uu−1 ) det e + 2tP ( u)α
√
= det e + 2tP ( u)α
√
= det e + 2tP ( α)u .
The last equality follows again from Proposition A.2.2 (iv), which implies
√
√
det(P ( u)α) = det(u) det(α) = det(P ( α)u).
Hence φ(t, u) can be written as
φ(t, u) =
δ
δ
ln det(u) + ln det(u−1 + 2tα).
2
2
(3.43)
Using expression (3.43) for φ(t, u) and ∇ ln det x = x−1 yields
δ −1
∂φ(t, u)
−1
=
(u + 2tα) , 2α = hδα, ψ(t, u)i,
∂t
2
and shows that (3.41) solves (3.39).
Let now ψ(t, u) and φ(t, u) be given by (3.38) and (3.39) and consider
− δ
√
−1
−1
Lδ,α,x
(u) := e−φ(t,u)−hψ(t,u),xi = det e + 2tP ( α)u 2 e−h(u +2tα) ,xi
t
for u ∈ K̊. We shall now prove that for every t ∈ R+ and α, x ∈ K
u 7→ Lδ,α,x
(u)
t
(3.44)
is the Laplace transform of a probability measure on K if δ ≥ d(r − 1).
In the case K = Sr+ , this is implied by Letac and Massam [2004, Proposition
88
Chapter 3. Affine Processes on Symmetric Cones
3.2], which asserts that Lδ,α,x
corresponds to the Laplace transform of the nont
central Wishart distribution. The proof is based on the density function of this
distribution, which exists for δ > (r − 1) and α ∈ Sr++ . As such a result is not
available for general symmetric cone, we choose a different approach. However,
in Section 3.5 we establish the form of the density corresponding to Lδ,α,x
,
t
which then yields a generalization of the non-central Wishart distribution on
symmetric cones.
In order to prove (3.44), the following lemma states that it is enough to
consider α = e and t = 12 .
Lemma 3.4.2. Let α ∈ K and δ ≥ d(r − 1) and t ∈ R+ be fixed. Suppose
that for every x ∈ K
δ
−1 +e)−1 ,xi
Lδ,e,x
(u) = det (e + u)− 2 e−h(u
1
2
is the Laplace transform of a probability measure on K. Then this assertion
also holds true for Lδ,α,x
.
t
Proof. Let t > 0 and x be fixed. Suppose first that α ∈ K̊. We denote by X
. Consider now
the random variable whose Laplace transform is given by Lδ,e,x
1
2
√
√
another random variable Y defined by Y = 2tP ( α)X and let y = 2tP ( α)x.
We then have by Proposition A.2.2 (i),
h
i
√
E e−hu,Y i = E e−h2tP ( α)u,Xi
√
− δ
√
−1
−1
= det e + 2tP ( α)u 2 e−h((2tP ( α)u) +e) ,xi
√
√
√
− δ
√
−1
−1 1
−1
= det e + 2tP ( α)u 2 e−h2tP ( α)(u +2tP ( α)e) , 2t P ( α )yi
− δ
√
−1
−1
= det e + 2tP ( α)u 2 e−h(u +2tα) ,yi
= Lδ,α,y
.
t
√
Since x was arbitrary and as P ( α) is an automorphism on K, Lδ,α,y
is a
t
Laplace transform of a probability distribution for every y ∈ K. For degenerate α the assertion is a consequence of Lévy’s continuity
theorem, since
e
,x
δ,α+ n
δ,α,x
to Lt , which is a
we have locally uniform convergence on K̊ of Lt
continuous function at 0.
δ
Due to this lemma, it is sufficient to consider det (e + u)− 2 e−h(u +e) ,xi .
In the following we fix a Jordan frame p1 , . . . , pr and denote for all m ∈
{1, . . . , r}
m
X
em :=
pi , V (m) := V (em , 1),
i=1
−1
−1
3.4. Construction of Affine Processes on Symmetric Cones
89
where V (em , 1) = {x ∈ V : em ◦ x = x} (see Section A.1.2). Moreover,
K (m) corresponds to the symmetric cone associated with the subalgebra V (m)
of rank m, that is, the relative interior of the set of squares in V (m) . For a
(m)
general indempotent c ∈ V (m) , Kc denotes the symmetric cone associated
with the subalgebra V (m) (c, 0).
The following lemmas are based on an idea of Ahdida and Alfonsi [2010].
Lemma 3.4.3. Let m ∈ {1, . . . , r}. Fix δ ≥ d(m − 1), t ∈ R+ and x ∈
δ,e
,x
m ,x
K (m) . Suppose that Lt m−1 and Lδ,p
are Laplace transforms of probability
t
m ,x
(m)
measures on K . Then the same holds true for Lδ,e
.
t
Proof. Define φi (t, u) and ψi (t, u) for i = 1, 2 and u ∈ K̊ by
δ
ln det(e + 2tP (pm )u),
2
δ
φ2 (t, u) = ln det (e + 2tP (em−1 )u) ,
2
ψ1 (t, u) = (u−1 + 2tpm )−1 ,
φ1 (t, u) =
ψ2 (t, u) = u−1 + 2tem−1
δ,em−1 ,x
m ,x
Hence Lδ,p
= e−φ1 (t,u)−hψ1 (t,u),xi and Lt
t
claim that
−1
= e−φ2 (t,u)−hψ2 (t,u),xi . We now
δ
−1 +2te
m ,x
Lδ,e
(u) = e−φ(t,u)+hψ(t,u),xi = det (e + 2tP (em )u)− 2 e−h(u
t
−1 ,xi
m)
is given by e−φ1 (t,u)−φ2 (t,ψ1 (t,u))−hψ2 (t,ψ1 (t,u)),xi . Indeed, we have
ψ2 (t, ψ1 (t, u)) =
−1
u
+ 2tpm
−1 −1
−1
+ 2tem−1
−1
= u−1 + 2tpm + 2tem−1
= (u−1 + 2tem )−1 = ψ(t, u),
and by using (3.42),
e−φ1 (t,u)−φ2 (t,ψ1 (t,u))
δ
= det(e + 2tP (pm )u)− 2 det e + 2tP (em−1 ) (u−1 + 2tpm )−1
δ
δ
δ
δ
δ
= det(u)− 2 det(u−1 + 2tpm )− 2 det((u−1 + 2tpm )−1 )− 2
− δ
× det ((u−1 + 2tpm )−1 )−1 + 2tem−1 2
= det(u)− 2 det(u−1 + 2tem )− 2
δ
= det (e + 2tP (em )u)− 2 = e−φ(t,u) .
.
− 2δ
90
Chapter 3. Affine Processes on Symmetric Cones
δ,em−1 ,x
m ,x
Denoting the probability measures associated with Lδ,p
and Lt
t
2
1
pt (x, dξ) and pt (x, dξ), we thus have
by
m ,x
Lδ,e
(u) = e−φ1 (t,u)−φ2 (t,ψ1 (t,u))−hψ2 (t,ψ1 (t,u)),xi
t
Z
Z
e
e 2 (x, dξ),
=
e−hu,ξi p1t (ξ, dξ)p
t
K (m)
K (m)
m ,x
implying that Lδ,e
is the Laplace transform of the probability measure
t
Z
p1t (ξ, ·)p2t (x, dξ).
K (m)
The following lemma states a decomposition of x into the sum of two
elements of smaller rank.
Lemma 3.4.4. Let x ∈ K̊ (m) for some m ∈ {2, . . . , r} and c an idempotent in
V (m) of rank k, where k ∈ {1, . . . , m − 1}. Consider the Peirce decomposition
of x in V (m) with respect to c, that is, x = x1 + x 1 + x0 and denote the inverse
2
3
of x1 in V (m) (c, 1) by x−1
1 . Then x can be decomposed into x = y + z, where
(m)
,
y = x1 + x 1 + P (x 1 )x−1
1 ∈ K
(3.45)
(m)
z = x0 − P (x 1 )x−1
⊂ V (m) (c, 0).
1 ∈ Kc
(3.46)
2
2
2
Proof. By the Peirce multiplication rule (A.6), we have
1
(m)
V (m) (c, 1) ⊂ V (m) (c, 0),
P V
c,
2
(m)
whence P (x 1 )x−1
∈ Kc , from which we can deduce that y ∈ K (m) . Us1
2
ing Massam and Neher [1997, Proposition 3.3.1 c)], we can write
((x−1 )0 )−1 = x0 − P (x 1 )x−1
1 ,
2
(m)
where ((x−1 )0 )−1 is the inverse of (x−1 )0 in V (m) (c, 0). As this lies in Kc ,
the assertion is proved.
Remark 3.4.5.
3
(i) In the case V (m) = Sm , the matrix
Ik 0
0 0
Note that this is well-defined, since we assume x ∈ K̊ (m) .
3.4. Construction of Affine Processes on Symmetric Cones
91
is an idempotent of rank k. The above decomposition then corresponds
to
x1 x>
x1
x>
0
0
12
12
=
+
.
>
>
x12 x0
x12 x12 x−1
0 x0 − x12 x−1
1 x12
1 x12
(ii) In the case of the two-dimensional Lorentz cone with m = r = 2 and
k = 1, the Peirce decomposition with respect to an idempotent of rank
1 is given by x = x1 + x0 , since dim(V (c, 12 )) = 0. Thus the above
decomposition reduces to y = x1 and z = x0 .
In the following we show that, for integer values k and particular elements
m −c,y
y of form (3.45), Ldk,e
can be recognized as the Laplace transform of
1
2
a quadratic function of a Gaussian random variable supported on V (c, 21 ),
where c denotes some idempotent specified below. Before stating the lemma
for general irreducible symmetric cones, let us examplify the assertion for
+
such that its Peirce decomposition with respect to
V (m) = Sm . Let y ∈ Sm
Ik 0
0 0
is given by
>
y1
y12
.
>
y12 y12 y1−1 y12
Consider for some z ∈ V (m) (Ik , 21 ) the following linear transformation of symmetric matrices
x1
x1 z > + x>
x1 x>
12
12
.
7→
τ (z) : Sm → Sm ,
x12 x0
zx1 + x12 zx1 z > + x12 z > + zx>
12 + x0
Let now Z be a standard Gaussian random variable on V (m) (Ik , 12 ). We then
+
prove that the Laplace transform of the Sm
-valued random variable
q
√
>
y1
y1 Z > + y12
−1
√
τ 2 y1 ◦ Z y =
Z y1 + y12
q(Z)
with
q q >
−1
q(Z) = Z + y12 y1
Z + y12 y1−1
m −Ik ,y
is given by Lk,I
, where we identify Z with an element in Rm−k×k .
1
2
The generalization of the map τ (z) to Euclidean Jordan algebras is the
so-called Frobenius transform introduced in Definition A.2.3. This transform
is used to prove the following lemma.
92
Chapter 3. Affine Processes on Symmetric Cones
Lemma 3.4.6. Let c be an idempotent in V (m) of rank k and tr(c) = k,
where m ∈ {2, . . . , r} and k ∈ {1, . . . , m − 1}. Moreover, let y ∈ K (m) be of
form (3.45), that is,
y = y1 + y 1 + y0 = y1 + y 1 + P (y 1 )y1−1
2
2
2
for some y1 ∈ V (m) (c, 1) and y 1 ∈ V (m) (c, 12 ). Then
2
dk
−1 +(e
m −c,y
u 7→ Ldk,e
(u) = det(e + P (em − c)u)− 2 e−h(u
1
−1 ,yi
m −c))
,
u ∈ K̊,
2
(3.47)
is the Laplace transform of a probability distribution on K (m) .
Proof. We show that there exists a random variable X taking values in K (m)
such that
dk
−1
−1
E e−hu,Xi = det(e + P (em − c)u)− 2 e−h(u +(em −c)) ,yi .
By Faraut and Korányi [1994, Proposition VI.4.1], the function
σ:V
(m)
1
(m)
(em − c, 1) → L V
,
c,
2
defined by
σ(x)ξ = 2x ◦ ξ = 2L(x)ξ
is a self-adjoint representation of V (m) (em − c, 1) on V (m) (c, 12 ) with
σ(em − c) = I,
where I denotes the identity. In particular, since V (m) (em − c, 1) = V (m) (c, 0),
(m)
(m)
there is a Kc -valued quadratic form Q : V (m) (c, 21 ) → Kc ⊂ V (m) (c, 0)
given by
hσ(x)ξ, ξi = h2x ◦ ξ, ξi = hx, 2ξ 2 i = hx, Q(ξ)i,
x ∈ V (m) (c, 0),
which implies that Q(ξ) = 2L(em − c)ξ 2 . Let now Z be a Gaussian random
variable Z ∼ N (0, I) on V (m) (c, 12 ) and define the K (m) -valued random variable
X by
q
−1
X = τ 2 y1 ◦ Z y,
3.4. Construction of Affine Processes on Symmetric Cones
93
where τ denotes the Frobenius transform, as introduced in Definition A.2.3.
According to Faraut and Korányi [1994, Lemma VI.3.1], the Peirce decomposition of X with respect to c is then given by
X 1 = y1 ,
q
−1
X 1 = 4L
y1 ◦ Z y1 + y 1 ,
2
2
q
2
q
−1
−1
X0 = 2L(em − c)L 2 y1 ◦ Z y1 + 4L(em − c)L
y1 ◦ Z y 1 + y0 .
2
Denoting by L(em )u = u1 + u 1 + u0 the Peirce decomposition of L(em )u
2
with respect to c in V (m) , we obtain due to the orthogonality of V (c, λ), for
λ = {1, 21 , 0} and the fact that X takes values in V (m)
q
−1
hu, Xi = hu1 , y1 i + u 1 , 4L
y1 ◦ Z y1 + y 1
2
2
*
q
q
2
+
u0 , 2L 2 y1−1 ◦ Z
y1 + 4L
+
y1−1 ◦ Z y 1 + y0
2
,
E
D
(3.48)
= hu1 , y1 i + u 1 , y 1 + hu0 , y0 i
2
2
q q −1
−1
+4 L
y1
y1
L(y1 )u 1 , Z + 4 L
L(y 1 )u0 , Z
2
+
2
1
hu0 , Q(Z)i .
2
Here, the last equality follows from
q
2
1
−1
2L(em − c)L 2 y1 ◦ Z y1 = L(em − c)Z 2 = Q(Z),
2
which is a consequence of Massam and Neher [1997, Equation 3.5, 3.6].
For notational reasons we define
q q 1 √
−1
η := L
y1
L(y1 )u 1 + L(y 1 )u0 = L u 1
y1 + L(u0 )L
y1−1 y 1 ,
2
2
2
2
2
where we use Faraut and Korányi [1994, Proposition II.1.1] and again Massam
and Neher [1997, Equation 3.5]. Let us now compute
h 1
i
E e− 2 hu0 ,Q(Z)i−4hη,Zi .
94
Chapter 3. Affine Processes on Symmetric Cones
Denoting the dimension of V (m) (c, 21 ) by N := dk(m − k) and using
σ(em − c) = I, we have
h 1
i
− 2 hu0 ,Q(Z)i−4hη,Zi
E e
Z
1
− 12 hu0 ,Q(z)i−4hη,zi − 12 hσ(em −c)z,zi
=
e
dz
N e
1
V (m) (c, 2 ) (2π) 2
1
= Det(σ(u0 + (em − c)))− 2
Z
1
1
− 21 h(σ(u0 +em −c)z,zi−4hη,zi
×
Det(σ(u0 + (em − c))) 2 dz
N e
V (m) (c, 12 ) (2π) 2
N
1
= det(u0 + (em − c))− 2(m−k) e 2 hσ((u0 +em −c)
dk
= det(u0 + (em − c))− 2 e16hL((u0 +(em −c))
−1 )4η,4ηi
−1 )η,ηi
.
Here, we use the moment generating function for the normal distribution with
N
covariance σ((u0 +(em −c))−1 ), the relation Det(σ(x)) = det(x) m−k (see Faraut
and Korányi [1994, Proposition IV.4.2]) and the fact that σ(x) = 2L(x) in our
case. The linear map L is restricted to V (m) (c, 0) = V (m) (em − c, 1) such that
the inverse is well-defined. Note that det(u0 + (em − c)) = det(P (em − c)u +
(em − c)) = det(P (em − c)u + e). Due to (3.48), it thus only remains to prove
− hu1 , y1 i − hu 1 , y 1 i − hu0 , y0 i + 16hL((u0 + (em − c))−1 )η, ηi
2
2
= −h(u−1 + (em − c))−1 , yi. (3.49)
By the definition of η and the fact that
y0 =
P (y 1 )y1−1
2
q
2
−1
= L(em − c)L 2 y1 ◦ y 1 e,
2
we obtain using several times Massam and Neher [1997, Equation 3.5]
− hu1 , y1 i − hu 1 , y 1 i − hu0 , y0 i + 16hL((u0 + (em − c))−1 )η, ηi
2
2
= h2L(u 1 )L(u 1 )(u0 + (em − c))−1 − u1 , y1 i
2
2
+ h2L(u )L(u0 )(u0 + (em − c))−1 − u 1 , y 1 i
1
2
2
−1
+ hL(u0 )L(u0 )(u0 + (em − c))
2
− u0 , y0 i.
On the other hand we can express −L(em )(u−1 + (em − c))−1 as
− L(em )(u−1 + (em − c))−1 = P (u0 + u 1 )(u0 + (em − c))−1 − L(em )u
2
= (P (u 1 ) + 2P (u0 , u 1 ) + P (u0 ))(u0 + (em − c))−1 − L(em )u.
2
2
3.4. Construction of Affine Processes on Symmetric Cones
95
The above claim (3.49) is then a consequence of the following identities (Massam and Neher [1997, Equation 3.7] for the first one)
P (u 1 )(u0 + (em − c))−1 = 2L(c)L(u 1 )L(u 1 )(u0 + (em − c))−1 ,
2
2
2
P (u0 , u 1 )(u0 + (em − c))−1 = L(u 1 )L(u0 )(u0 + (em − c))−1 ,
2
2
P (u0 )(u0 + (em − c))−1 = L(u0 )L(u0 )(u0 + (em − c))−1 ,
whence the assertion is proved.
Remark 3.4.7. Note that in the case of the two-dimensional Lorentz cone
y = y1 and (3.47) is reduced to e−hu1 ,y1 i , which is simply the Laplace transform
of the Dirac measure at y1 .
Using the above lemmas, we are now prepared to prove the crucial proposition which shall allow us to establish the existence of affine diffusion processes.
Proposition 3.4.8. Let m ∈ {1, . . . , r}. Then, for all δ0 ≥ 0 and x ∈ K (m) ,
δ +d(m−1),em ,x
u 7→ L 10
(u),
u ∈ K̊,
2
is the Laplace transform of a probability measure on K (m) .
Proof. We proceed by induction on m. Let us start with m = 1. Then, for all
δ0 ≥ 0 and x ∈ K (1) = R+ p1 , Lδ10 ,p1 ,x is a Laplace transform of a probability
2
distribution on K (1) = R+ p1 . Indeed,
δ0 ,p1 ,x
L1
2
−
δ0
2
E
D
−1
− (u−1 +p1 ) ,x
e
(u) = det (e + P (p1 )u)
1
sa
=
−
,
δ0 exp
1+s
(1 + s) 2
where s ≥ 0 is defined through P (p1 )u = sp1 and a ≥ 0 by x = ap1 . Here, we
use the fact that P (p1 ) is the orthogonal projection on V (1) .
Noticing that
sa
1
s 7→
−
(3.50)
δ0 exp
1+s
(1 + s) 2
is the Laplace transform of the image of the non-central chi-square distribution
(with δ0 degrees of freedom and non-centrality parameter depending on a)
under some positive linear map (see, e.g., Letac and Massam [2004, Definition
1.1]), implies the above claim.
96
Chapter 3. Affine Processes on Symmetric Cones
As induction hypothesis we now assume that for all δ0 ≥ 0 and z ∈ K (m−1) ,
L1
is a Laplace transform of a probability distribution on K (m−1) .
δ0 +d(m−2),em−1 ,z
2
For the induction step from m − 1 to m, we prove that for all x ∈ K (m) ,
δ +d(m−1),em−1 ,x
L 10
δ +d(m−1),pm ,x
and L 10
2
2
are Laplace transforms of probability distributions on K (m) . An application
of Lemma 3.4.3 then yields the assertion.
Let now x ∈ K̊ (m) be fixed. By Lemma 3.4.4, we can decompose x with
(m)
respect to pm into x = y +z, where y is of form (3.45) and z ∈ Kpm = K (m−1) .
δ +d(m−1),em−1 ,x
, which can therefore be written as
We start by considering L 10
2
δ +d(m−1),em−1 ,x
L 10
D
−1
D
−1
= det (e + P (em−1 ) u)−
δ0 +d(m−1)
2
− u−1 +em−1 )
e (
= det (e + P (em−1 ) u)−
δ0 +d(m−2)
2
− u−1 +em−1 )
e (
,x
E
2
D
d
− u−1 +em−1 )
× det (e + P (em−1 ) u)− 2 e (
−1
,y
,z
E
(3.51)
E
.
(3.52)
By our induction hypothesis, the first term (3.51) is the Laplace transform of
a probability distribution on K (m−1) . Due to Lemma 3.4.6, the same holds
true for the second term (3.52), but with support on K (m) . Hence the product
of these two terms is again the Laplace transform of a probability distribution
δ +d(m−1),pm ,x
, which by Lemma 3.4.4 can be
on K (m) . Let us now turn to L 10
2
written as
δ +d(m−1),pm ,x
L 10
= det(e + P (pm )u)−
δ0 +d(m−1)
2
= det(e + P (pm )u)−
d(m−1)
2
−1 +p
e−h(u
−1 ,xi
m)
2
δ
− 20
× det(e + P (pm )u)
−1 +p
e−h(u
−1 ,yi
m)
−h(u−1 +pm )−1 ,zi
e
,
(3.53)
(3.54)
(m)
where z ∈ Kem−1 , which is the cone associated with V (em−1 , 0), and y is of
form (3.45). Note that the corresponding idempotent is now em−1 . Due to
Lemma 3.4.6, the first term (3.53) corresponds again to the Laplace transform
of a probability distribution on K (m) , and the second one (3.54) to the non(m)
central chi-square distribution supported on the one-dimensional cone Kem−1 ,
similar as in (3.50), where p1 is simply replaced by pm . Hence, by Lemma 3.4.3,
e
δ0 +d(m−1),em ,x+ n
the assertion is proved for all x ∈ K̊ (m) . For degenerate x, L 1
2
δ +d(m−1),em ,x
converges locally uniformly on K̊ to L 10
. Due to the continuity
2
δ +d(m−1),em ,x
of u 7→ L 10
2
holds for x ∈ ∂ K̊ (m) .
(u) at 0 and Lévy’s continuity theorem, the claim also
3.4. Construction of Affine Processes on Symmetric Cones
97
The following corollary is an immediate consequence of the above proposition.
Corollary 3.4.9. Let ψ(t, u) and φ(t, u) be given by (3.38) and (3.39) and
consider
− δ
√
−1
−1
Lδ,α,x
(u) = e−φ(t,u)+hψ(t,u),xi = det e + 2tP ( α)u 2 e−h(u +2tα) ,xi
t
for u ∈ K̊. If δ ≥ d(r −1), then, for every t ∈ R+ and α, x ∈ K, u 7→ Lδ,α,x
(u)
t
is the Laplace transform of a probability measure on K.
Proof. From Proposition 3.4.8 it immediately follows that, for every δ ≥
d(r − 1) and x ∈ K, Lδ,e,x
is the Laplace transform of a probability measure
1
2
on K. An application of Lemma 3.4.2 then yields the assertion for general
parameters.
Using the knowledge that Lδ,α,x
is the Laplace transform of a probability
t
distribution on K, we can finally prove existence of affine diffusion processes
associated with the particular parameter set (α, δα, 0, 0, 0, 0, 0), where α ∈ K
and δ ≥ d(r − 1).
Proposition 3.4.10. Let (α, δα, 0, 0, 0, 0, 0) be an admissible parameter set,
that is, α ∈ K and δ ≥ d(r − 1). Then there exists a unique affine process on
K such that (2.1) holds for all (t, u) ∈ R+ × K, where φ(t, u) and ψ(t, u) are
given in Lemma 3.4.1.
Proof. By Proposition 3.4.9 we have for every (t, x) ∈ R+ × K the existence
of a probability measure on K with Laplace-transform e−φ(t,u)−hψ(t,u),xi , where
φ(t, u) and ψ(t, u) are specified in Lemma 3.4.1. The Chapman-Kolmogorov
equations hold in view of the flow property of φ and ψ, whence the assertion
follows.
Remark 3.4.11. In the case of positive semidefinite matrices, Bru [1991] has
shown existence and uniqueness for the process
p
p
dXt = δIr + Xt dWt + dW > Xt , X0 = x,
if δ > r − 1 and x with distinct eigenvalues (see Bru [1991, Theorem 2 and
Section 3]). This process corresponds to the parameter set (Ir , δIr , 0, 0, 0, 0, 0)
on the cone Sr+ (see Theorem 3.7.2 below). Note that, since the Peirce invariant d equals 1 in this case, δ > r − 1 is a stronger assumption than what
we require on δ. Actually, Bru [1991] establishes existence and uniqueness of
solutions also for δ = 1, . . . , r − 1. But these are degenerate solutions, as they
are only defined on lower dimensional subsets of the boundary of Sr+ (see Bru
[1991, Corollary 1]).
98
Chapter 3. Affine Processes on Symmetric Cones
3.4.2
Existence of Affine processes on Symmetric cones
In Proposition 3.4.10 and Proposition 2.5.3 we have proved existence of affine
diffusion processes with a particular constant drift parameter and existence of
pure affine jump processes. In order to establish existence of affine processes
on symmetric cones for any admissible parameter set, we now combine the
respective Riccati equations to show that
iN
h
1
2
−φ(t,u)−hψ(t,u),xi
e−hu,xi ,
e
= lim P t P t
N →∞
N
N
is the Laplace transform of a probability distribution on K for any admissible
parameter set. Here, P i , i = 1, 2, denote the respective semigroups of the
diffusion process and the pure jump process.
Given an admissible parameter set (α, b, B, c, γ, m, µ), let us therefore consider the following two systems of Riccati ODEs:
∂t ψ1 (t, u) = R1 (ψ1 (t, u)) = −2P (ψ1 (t, u))α,
∂t φ1 (t, u) = F1 (ψ1 (t, u)) = hδα, ψ1 (t, u)i,
∂t ψ2 (t, u) = R2 (ψ2 (t, u)) = B > (ψ2 (t, u)) + γ
Z
−
e−hξ,ψ2 (t,u)i − 1 + hχ(ξ), ψ2 (t, u)i µ(dξ),
K
Z
∂t φ2 (t, u) = F2 (ψ2 (t, u)) = hb − δα, ψ2 (t, u)i + c −
e−hξ,ψ2 (t,u)i − 1 m(dξ),
K
(3.55)
where we set δ = d(r − 1). The original Riccati equations corresponding to
the parameter set (α, b, B, c, γ, m, µ) are then given by
∂t ψ(t, u) = R(ψ(t, u)) = R1 (ψ(t, u)) + R2 (ψ(t, u)),
∂t φ(t, u) = F (ψ(t, u)) = F1 (ψ(t, u)) + F2 (ψ(t, u)),
ψ(0, u) = u,
φ(0, u) = 0.
(3.56)
(3.57)
Let us remark that, due to Theorem 2.4.3, there exists a global unique
solution to (3.56)-(3.57) for every u ∈ K̊, which remains in K̊ for all t ∈ R+ .
Lemma 3.4.12. Let φi , ψi , i = 1, 2, be defined by (3.55) and let u ∈ K̊ and
t ≥ 0 be fixed. Define recursively for each N ∈ N and n ∈ {0, . . . , N }
y0 (u) := u,
yn (u) := ψ2 (τ, ψ1 (τ, yn−1 )),
where τ =
t
.
N
w0 (u) := 0,
wn (u) := φ1 (τ, yn−1 ) + φ2 (τ, ψ1 (τ, yn−1 ) + wn−1 ,
Then
ψ(t, u) = lim yN (u)
N →∞
and
where φ and ψ are given by (3.56)-(3.57).
φ(t, u) = lim wN (u),
N →∞
3.4. Construction of Affine Processes on Symmetric Cones
99
Proof. Let us first remark that the limits are well-defined, since we have existence of global solutions of (3.56)-(3.57) by Theorem 2.4.3. In order to prove
convergence of this splitting scheme, let us first calculate the local errors of the
approximations for φ and ψ for a given step size τ = Nt with N ∈ N fixed. An
estimate of the global error is then obtained by transporting the local errors
to the final point t and adding them up, as it is done in Hairer, Nørsett, and
Wanner [1993, Theorem 3.6]. Following Hairer et al. [1993, Chapter II.3], let
us define the increment functions Φψ and Φφ by
yn (u) = yn−1 (u) + τ Φψ (yn−1 (u), τ ),
wn (u) = wn−1 (u) + τ Φψ (yn−1 (u), τ ).
Using a Taylor expansions at τ = 0, we obtain due to the analyticity of R1 , R2
and F1 , F2 on K̊ for y ∈ K̊
Φψ (y, τ ) = R2 (y) + R1 (y)
1
+ τ (DR2 (y)R2 (y) + 2DR2 (y)R1 (y) + DR1 (y)R1 (y))
2
+ O(τ 2 ),
Φφ (y, τ ) = F1 (y) + F2 (y)
1
+ τ (hDF2 (y), R2 (y)i + 2hDF2 (y), R1 (y)i + hDF1 (y), R1 (y)i)
2
+ O(τ 2 ).
Hence the local errors satisfy by another Taylor expansion of ψ and φ
kψ(t + τ, u) − ψ(t, u) − τ Φψ (ψ(t, u), τ )k
1
= τ 2 kDR1 (ψ(t, u))R2 (ψ(t, u)) − DR2 (ψ(t, u))R1 (ψ(t, u))k + O(τ 3 )
2
≤ Cψ τ 2 ,
|φ(t + τ, u) − φ(t, u) − τ Φφ (ψ(t, u), τ )|
1
= τ 2 |hDF1 (ψ(t, u)), R2 (ψ(t, u))i − hDF2 (ψ(t, u)), R1 (ψ(t, u))i| + O(τ 3 )
2
≤ Cφ τ 2 .
Since R1 , R2 and F1 , F2 are analytic on K̊, the following Lipschitz conditions
for some constants Λψ , Λφ are satisfied in a neighborhood of the solution
kΦψ (z, τ ) − Φψ (y, τ )k ≤ Λψ kz − yk,
|Φφ (z, τ ) − Φφ (y, τ )| ≤ Λφ kz − yk.
100
Chapter 3. Affine Processes on Symmetric Cones
Moreover, by Theorem 2.4.3, ψ(t, u) ∈ K̊ for all (t, u) ∈ R+ × K̊ such that we
have by Hairer et al. [1993, Theorem II.3.6]
Cψ Λψ t−1
e
,
Λψ
Cφ
kφ(t, u) − wN (u)k ≤ τ eΛφ t−1 ,
Λφ
kψ(t, u) − yN (u)k ≤ τ
which converges by the definition of τ to 0 as N → ∞.
We are now prepared to prove the main result of this section, which establishes existence of affine processes on irreducible symmetric cones for any
given admissible parameter set.
Theorem 3.4.13. Let (α, b, B, c, γ, m, µ) be an admissible parameter set. Then
there exists a unique affine process on K, such that (2.1) holds for all (t, u) ∈
R+ × K, where φ(t, u) and ψ(t, u) are given by (2.9) and (2.10).
Proof. By Lemma 3.4.12, we have for each fixed t
e−φ(t,u)−hψ(t,u),xi = lim e−wN (u)−hyN (u),xi ,
N →∞
u ∈ K̊.
For each N ∈ N, n ∈ {0, . . . , N } and x ∈ K, u 7→ e−wn (u)−hyn (u),xi is the
Laplace transform of a probability distribution on K. Indeed, let us proceed
by induction. For n = 0, e−hu,xi is the Laplace transform of δx (dξ). We
now suppose that for every x ∈ K, e−wn−1 −hyn−1 ,xi is the Laplace transform
of a probability distribution µn−1 (x, ·) on K. Due to Proposition 3.4.9 and
Proposition 2.5.2, u 7→ e−φi (τ,u)−hψi (τ,u),xi , i = 1, 2, are Laplace transforms of
probability measures supported on K, which we denote by piτ (x, dξ), i = 1, 2.
Since we have
e−wn −hyn ,xi = e−wn−1 e−φ1 (τ,yn−1 )−φ2 (τ,ψ1 (τ,yn−1 ))−hψ2 (τ,ψ1 (τ,yn−1 )),xi
Z Z
e
e 2 (x, dξ),
=
e−wn−1 −hyn−1 ,ξi p1τ (ξ, dξ)p
τ
ZK ZK Z
e dz)p1 (ξ, dξ)p
e 2 (x, dξ),
=
e−hu,zi µn−1 (ξ,
τ
τ
K
K
K
e−wn −hyn ,xi is the Laplace transform of the probability distribution given by
Z Z
e ·)p1 (ξ, dξ)p
e 2 (x, dξ).
µn (x, ·) =
µ(ξ,
τ
τ
K
K
3.5. Wishart Distribution
101
As u 7→ e−φ(t,u)−hψ(t,u),xi is continuous at 0 and the limit of a sequence of
Laplace transforms of probability distributions supported on K, Lévy’s continuity theorem implies that u 7→ e−φ(t,u)−hψ(t,u),xi is also the Laplace transform
of a probability distribution on K.
Moreover, the Chapman-Kolmogorov equation holds in view of the semiflow property of φ and ψ, which implies the assertion.
3.5
Wishart Distribution
In Proposition 3.4.9 we have seen that for every t ∈ R+ and α, x ∈ K
− δ
√
−1
−1
Lδ,α,x
(u) = det e + 2tP ( α)u 2 e−h(u +2tα) ,xi
t
is the Laplace transform of a probability measure on K if δ ≥ d(r − 1). In the
case of positive semidefinite matrices, this probability measure corresponds
actually to the non-central Wishart distribution (see, e.g., Letac and Massam
[2004]). For certain parameter values, namely δ > (r − 1) and α ∈ Sr++ ,
this distribution admits a density, which is explicitly known. We extend this
to general irreducible symmetric cones and establish the explicit form of the
Markov kernels corresponding to the affine diffusion processes, associated with
the parameter set (α, δα, 0, 0, 0, 0, 0), for δ > d(r − 1) and α ∈ K̊.
3.5.1
Central Wishart Distribution
We start by analyzing the case x = 0, which corresponds to the central Wishart
distribution (see, e.g., Letac and Massam [2004] or Massam and Neher [1997]).
The following proposition extends the assertion of Corollary 3.4.9 to the set
δ ∈ {0, d, . . . , d(r − 1)} and states the form of the density in the case δ >
d(r − 1) and α ∈ K̊.
Proposition 3.5.1. Let φ(t, u) be given by (3.39) and consider
− δ
√
−φ(t,u)
Lδ,α
= det e + 2tP ( α)u 2 .
t (u) := e
If δ belongs to the set
G = {0, d, . . . , d(r − 1)} ∪ ]d(r − 1), ∞[ ,
then, for every t ∈ R+ and α ∈ K, u 7→ Lδ,α
t (u) is the Laplace transform of a
δ
probability measure Wt2
,α
on K. Moreover, if δ > d(r − 1) and if α ∈ K̊, then
102
δ
Wt2
Chapter 3. Affine Processes on Symmetric Cones
,α
admits a density, which is given by
δ
,α
2
Wt
(ξ) =
1
ΓK
δ
2
det
α−1
2t
2δ
e
D −1 E
− α2t ,ξ
δ
n
det(ξ) 2 − r ,
(3.58)
where ΓK denotes the Gamma function of K (see Faraut and Korányi [1994,
Section VII.1]).
Proof. By Faraut and Korányi [1994, Theorem VII.3.1 and Proposition VII.2.3],
δ
det(u)− 2 is the Laplace transform of a positive measure if and only if δ ∈ G.
δ
This is equivalent to the fact that det(u)− 2 is a function of positive type (see,
e.g., Faraut and Korányi [1994, page 136]), that is,
N
X
δ
det(ui + uj )− 2 ci c̄j ≥ 0
i,j=1
for all choices of u1 , . . . , uN ∈ K and complex numbers c1 , . . . , cN . For every
√
− 2δ
t ∈ R+ , Lδ,α
is therefore also a function of positive
t (u) = det (e + 2tP ( α)u)
type and hence the Laplace transform of a positive measure if δ ∈ G. Since
δ,α
Lδ,α
t (u + v) ≤ Lt (u) for all u, v ∈ K, the measure is supported on K. As
δ,α
Lδ,α
t (u) is bounded and Lt (0) = 1, the measure is actually a probability
measure.
Concerning the second assertion, we have by Faraut and Korányi [1994,
Corollary VII.1.3] and Proposition A.2.2 (iv)
Z
D
E
−1
− u+ α2t ,ξ
e
K
det(ξ)
δ
−n
2
r
− 2δ
δ
α−1
dξ = ΓK
det u +
2
2t
!− δ
√
2
√
P ( α−1 )
δ
= ΓK
det
2tP ( α)u + e
2
2t
−1 − 2δ
− δ
√
δ
α
= ΓK
det
det e + 2tP ( α)u 2 .
2
2t
δ
The definition of the density of Wt2
,α
then yields the assertion.
Remark 3.5.2. (i) Analogous to the cone of positive semidefinite matrices,
one can define the central Wishart distribution W p,σ with shape parameter
d
d(r − 1)
d(r − 1)
e
p ∈ G = 0, , . . . ,
∪
,∞
2
2
2
3.5. Wishart Distribution
103
and scale parameter σ ∈ K on a symmetric cone by its Laplace transform
which takes the form stated in Proposition 3.5.1, that is,
Z
−p
√
e−hu,ξi W p,σ (dξ) = det e + P ( σ)u
.
K
(see, e.g., Massam and Neher [1997, Corollary 3]).
(ii) If p = k d2 , k ∈ {1, . . . , r − 1}, then the Wishart distribution is supported
on the set of elements in K which are precisly of rank k. This is a
consequence of Faraut and Korányi [1994, Proposition
In the
Pr VII.2.3].
>
+
case of Sr , this corresponds to the distribution of i=1 Zi Zi , where Zi
are standard Gaussian random variables in Rr .
3.5.2
Non-central Wishart distribution
In order to formulate Proposition 3.5.4 below, where we establish the form of
the density function of the non-central Wishart distribution on a symmetric
cone, let us introduce the so-called zonal polynomials (see Faraut and Korányi
[1994, Section XI.3, p. 234]).
Definition 3.5.3 (Zonal Polynomials). For each multi-index
m = (m1 , . . . , mr ) ∈ Nr
with length |m| := m1 + · · · + mr , we consider the generalized power function
defined by
∆m (ξ) = ∆1 (ξ)m1 −m2 ∆2 (ξ)m2 −m3 . . . ∆r (ξ)mr ,
where ∆i denotes the principal minors corresponding to the Jordan subalgebras
V (j) = V (p1 + . . . + pj , 1) with p1 , . . . , pr some fixed Jordan frame. The mth
zonal polynomial Zm is now defined by
Z
Zm (ξ) = ωm
∆m (Oξ)dO,
O∈O
where dO is the normalized Haar measure on O and O = G ∩ O(V ), where
G is the connected component of the identity in the automorphism group of
K and O(V ) the orthogonal group of V . Moreover, ωm denotes some positive
normalizing constant.
We now state for α ∈ K̊ and δ > d(r − 1) the form of the density whose
Laplace transform is given by Lδ,α,x
. To this end we generalize a result by Letac
t
and Massam [2004] on the density function of the non-central Wishart distribution on positive semidefinite matrices to symmetric cones.
104
Chapter 3. Affine Processes on Symmetric Cones
Proposition 3.5.4. Let ψ(t, u) and φ(t, u) be given by (3.38) and (3.39) and
consider
− 2δ −h(u−1 +2tα)−1 ,xi
√
−φ(t,u)+hψ(t,u),xi
e
Lδ,α,x
(u)
:=
e
=
det
e
+
2tP
(
α)u
t
for u ∈ K̊. If δ > d(r − 1) and if α ∈ K̊, then Lδ,α,x
(u) is the Laplace
t
transform of a density, which is given by
δ
,α,x
2
Wt
δ D
E
δ
n
α−1 2 − α2t−1 ,ξ+x
(ξ) = det
e
det(ξ) 2 − r
2t
!
√
X Zm 12 P ( x) P (α−1 ) ξ
4t
,
×
δ
|m|!Γ
m
+
K
2
m≥0
(3.59)
(3.60)
where ΓK denotes the Gamma function of K (see Faraut and Korányi [1994,
Section VII.1]) and Zm the zonal polynomials introduced in Definition 3.5.3.
Proof. We apply similar arguments as in Letac and Massam [2004] to prove the
δ
,α,x
assertion. We first establish that Wt2 , as given in (3.59), is a well-defined
positive measure. Concerning the convergence of
X Zm
m≥0
√
( x) P (α−1 ) ξ
,
|m|!ΓK m + 2δ
1
P
4t2
(3.61)
we can estimate ΓK (m+ 2δ ) due to Faraut and Korányi [1994, Theorem VII.1.1]
by
n−r
δ
ΓK m +
≥ (2π) 2 (min Γ(z))r =: M,
z≥0
2
where Γ denotes the Gamma function on R. This implies convergence of (3.61),
since we have by Faraut and Korányi [1994, Proposition XII.1.3(i)]
etr(ξ) =
X Zm (ξ)
|m|!
m≥0
(3.62)
for every ξ ∈ K. Due to the definition of the zonal polynomials, in particular
since ∆m (ξ) > 0 for all ξ ∈ K \ {0}, W δ ,α,x is therefore a well-defined positive
2
measure. Let us now prove that u 7→ Lδ,α,x
(u) is the Laplace transform of
t
δ
Wt2
,α,x
. For each m ∈ Nr and each automorphism g, we have by Faraut and
3.5. Wishart Distribution
105
Korányi [1994, Lemma XI.2.3] and Proposition A.2.2 (iv)
Z
e
E
D
−1
− u+ α2t ,ξ
δ
n
det(ξ) 2 − r Zm (gξ)dξ
K
− 2δ
−1 !
α−1
α−1
det u +
= ΓK
Zm g u +
2t
2t
−1 − 2δ
√ − 2δ
δ
α
= ΓK m +
det
α u
det e + 2tP
2
2t
−1 !
α−1
.
× Zm g u +
2t
δ
m+
2
(3.63)
√
If x is non-degenerate, P ( x)P (α−1 ) is an automorphism and plays the role
of g in our case. However, the above formula also holds true if x P
is degenerate.
Indeed, let us approximate x by xn = x + n1 e. Since (tr(ξ))k = |m|=k Zm (ξ)
for every k (see p. 235 in Faraut and Korányi [1994]), we have for |m| = k
k
k
√
Zm P ( xn ) P α−1 ξ ≤ tr P α−1 ξxn
≤ tr P α−1 ξx1
.
Dominated convergence then yields (3.63) also for degenerate x. By (3.62) we
obtain
Z
δ
√ 2δ
,α,x
α u
e−hu,ξi Wt2 dξ
det e + 2tP
K

√
1
−1
Z
P
(
x)
P
(α
)
u+
D −1 E
m
2
4t
X
− α2t ,x 

=e

|m|!
m≥0
−
=e
D
α−1
,x
2t
E +
1
P
4t2
√
( x)P (α−1 )
−1 −1
u+ α2t
,e
α−1
2t
−1 



(3.64)
.
√
√
Using P (z) = P ( z)P ( z), Proposition A.2.2 (ii) and Faraut and Korányi
[1994, Exercise II.5 (c)], which asserts (z+e)−1 −e = −(z −1 +e)−1 for invertible
106
Chapter 3. Affine Processes on Symmetric Cones
elements z, z + e and z −1 + e, we get
+
−1 *
−1 −1
√ 1
α
α
,x +
,e
−
P
x P α−1 u +
2t
4t2
2t
*
+
−1
1 √ −1 α−1
1 √ −1 = −e + P
α
u+
, P
α
x
2t
2t
2t
−1 1 √
√ −1
α u+e
, P
α
x
= −e + 2tP
2t
−1 1 √
√ −1
−1
=−
2tP
α u
+e
α
x
, P
2t
√ −1
√ −1 1 √ −1 = − 2tP
α u + 2tP
α e
, P
α
x
2t
= − (u−1 + 2tα)−1 , x
= − hψ(t, u), xi .
This proves that u 7→ Lδ,α,x
(u) is the Laplace transform of the density given
t
in (3.59).
Remark 3.5.5. The explicit form of the Markov kernels corresponding to the
affine diffusion processes associated with the parameter set (α, δα, 0, 0, 0, 0, 0),
δ
for δ > d(r − 1) and α ∈ K̊, is thus given by Wt2
3.6
,α,x
.
Relation to Infinitely Divisible Distributions
Let P be the set of all families of probability measures (Px )x∈K on the canonical
probability space (Ω, F) such that (X, (Px )x∈K ) is a stochastically continuous
Markov processes on K with Px [X0 = x] = 1 for all x ∈ K. For two probability
measures P, Q on (Ω, F) the convolution P∗Q is defined as the push-forward of
the product measure P×Q under the map (ω, ω 0 ) 7→ ω +ω 0 : (Ω×Ω, F ⊗F) →
(Ω, F).
Definition 3.6.1. An element (Px )x∈K ∈ P is called
(k)
(i) infinitely decomposable if, for each k ≥ 1, there exists (Px )x∈K ∈ P
such that
(k)
(k)
Px(1) +···+x(k) = Px(1) ∗ · · · ∗ Px(k) .
3.6. Relation to Infinitely Divisible Distributions
107
(ii) infinitely divisible if the one-dimensional marginal distributions Px ◦Xt−1
are infinitely divisible for all (t, x) ∈ R+ × K.
In Duffie et al. [2003, Theorem 2.15] it was shown that affine processes
n
on Rm
+ × R are infinitely decomposable Markov processes, and vice versa.
In fact, this property was the core for the existence proof of affine processes
in Duffie et al. [2003]. As we have seen in the previous section, the situation on
symmetric cones is different, since the non-central Wishart distributions are no
longer infinitely divisible (see, e.g., Lévy [1948] or Donati-Martin, Doumerc,
Matsumoto, and Yor [2004, Section 2.C] and also Remark 3.5.2). It turns out
that affine processes are infinitely decomposable Markov processes if and only
if α = 0 or the symmetric cone corresponds to R+ or to the two-dimensional
Lorentz cone. This is stated in Theorem 3.6.6 below.
We first prove some technical lemmas:
Lemma 3.6.2. Let g : K → R be an additive function, that is, g satisfies
Cauchy’s functional equation
g(x + y) = g(x) + g(y),
x, y ∈ K.
(3.65)
Then g can be extended to an additive function f : V → R. Moreover, if g is
measurable on K then f is measurable on V . In that case, f is a continuous
linear functional, that is, f (x) = hc, xi for some c ∈ V .
Proof. The first part follows from the fact that K is generating.
Concerning measurability, let E ∈ B(R) be a Borel measurable set. Then
we have by the additivity of f
f
−1
(E) =
=
=
∞
[
Bn =
n=1
∞
[
∞
[
{x + ne | x ∈ V, f (x) ∈ E, kxk ≤ n} − ne
n=1
{y ∈ V | f (y) ∈ E + f (ne), ky − nek ≤ n} − ne
n=1
∞
[
{y ∈ K | g(y) ∈ E + g(ne), ky − nek ≤ n} − ne,
n=1
which is, in view of the measurability of g on K, again a measurable set.
For x ∈ V we write x = (xi )i , where 1 ≤ i ≤ n. We introduce the
additive functions fi : R → R via fi (xi ) = f (0, . . . , 0, xi , 0, . . . , 0). By the
measurability of f , we infer that all fi are measurable functions on R. By Aczél
and Dhombres [1989, Chapter 2, Theorem 8], any additive measurable function
on the real line is a continuous linear functional. Hence, for each i,Pwe infer
the existence of ci ∈ R such that fi (xi ) = ci xi holds. Since f (x) = i fi (xi ),
it follows that f (x) = hc, xi for some c ∈ V .
108
Chapter 3. Affine Processes on Symmetric Cones
Let us now consider Cauchy’s exponential equation for h : K → R+ , that
is,
h(x + y) = h(x)h(y),
x, y ∈ K.
(3.66)
Lemma 3.6.3. Suppose h : K → R+ is measurable, strictly positive, and
satisfies (3.66). Then h(x) = e−hc,xi for some c ∈ V . If h ≤ 1, then c ∈ K.
Proof. Since h is strictly positive, its logarithm yields the well-defined function
g : K → R, g(x) := log h(x). Clearly g is additive, hence by the first part
of Lemma 3.6.2, there exists a unique additive extension f : V → R. Also,
f is measurable on K, hence by the second assertion of Lemma 3.6.2 we
have f (x) = −hc, xi for some c ∈ V . The last statement follows from the
monotonicity of the exponential and the self duality of K.
Remark 3.6.4. The assumption of strict positivity of h in the preceding
lemma is essential. Otherwise, there exist solutions h which are not of the
asserted form.
Lemma 3.6.3 is the main ingredient of the proof of the following characterization concerning k-fold convolutions of Markov processes:
(i)
Lemma 3.6.5. Let (Px )x∈K ∈ P (i = 0, 1, . . . , k). Then
(1)
(k)
Px(1) ∗ · · · ∗ Px(k) = P(0)
x ,
∀x(i) ∈ K,
x = x(1) + · · · + x(k) ,
(3.67)
(1)
(N )
) ∈
if and only if, for all t = (t1 , . . . , tN ) ∈ RN
+ and u = (u , . . . , u
N
(i)
(K)
,
N
∈
N,
there
exists
0
<
ρ
(t,
u)
≤
1
and
ψ(t,
u)
∈
K
such
that
Qk (i)
(0)
i=1 ρ (t, u) = ρ (t, u) and
h PN (i)
i
− i=1 hu ,Xti i
E(j)
e
= ρ(j) (t, u)e−hψ(t,u),xi , ∀ x ∈ K, j = 0, 1, . . . , k.
x
(3.68)
Proof. We proceed similarly as in the proof of Duffie et al. [2003, Lemma
10.3]. Fix k > 1, N > 1, t, u and set
i
h PN (i)
(j)
(j)
− i=1 hu ,Xti i
g (x) := Ex e
.
By the definition of the convolution, (3.67) is equivalent to the following
g (1) (x(1) ) · . . . · g (k) (x(k) ) = g (0) (x),
∀x(i) ∈ K,
x = x(1) + · · · + x(k) . (3.69)
Hence the implication (3.68) ⇒ (3.69) is obvious. For the converse direction
we observe that g (i) are strictly positive on all of K. Thus by (3.69) we have
g := g (1) /g (1) (0) = · · · = g (k) /g (k) (0) = g (0) /g (0) (0)
3.6. Relation to Infinitely Divisible Distributions
109
and g is a measurable, strictly positive function on K satisfying (3.66). Hence
an application of Lemma 3.6.3 yields the validity of equation (3.68), where
ρ(i) (t, u) = g (i) (0). By the definition of g (i) , it follows that 0 < ρ(i) (t, u) ≤ 1
and ψ(t, u) ∈ K.
Theorem 3.6.6. Let (Px )x∈K ∈ P and suppose that the rank and the Peirce
invariant of the Euclidean Jordan algebra V satisfy r > 1 and d > 0. Then
the following assertions are equivalent:
(i) (Px )x∈K is infinitely decomposable.
(ii) (X, (Px )x∈K ) is affine with vanishing diffusion parameter α = 0.
(iii) (X, (Px )x∈K ) is affine and infinitely divisible.
Proof. (i)⇒(ii): Due to Lemma 3.6.5, infinite decomposability implies that X
is affine. Also, by the definition of infinite decomposability and by Lemma 3.6.5
(k)
we have that the k th root (Px ) for each k ≥ 1 is an affine process with state
(k)
space K and exponents ψ(t, u) and φ(t, u)/k. This implies that (Px )x∈K
has admissible parameters (α, b/k, B, c/k, γ, m/k, µ). Hence the admissibility
condition proved in Proposition 3.2.6 implies b/k d(r − 1)α 0, for each k,
which is impossible, unless α = 0.
(ii)⇒(iii): This implication follows from Proposition 2.5.2, in view of the
Lévy-Khintchine form of −φ(t, ·) − hψ(t, ·), xi, for each t > 0.
(iii)⇒(i): By assumption, every transition kernel pt (x, dξ) of X is infinitely
divisible with Laplace transform Pt e−hu,xi = e−φ(t,u)−hx,ψ(t,u)i . For each k ≥ 1,
the maps φ(k) := φk , ψ (k) := ψ satisfy the properties (2.3)–(2.4). Moreover,
infinite divisibility implies that for each (t, x) ∈ R+ × K
(k)
Qt e−hu,xi := e−φ
(k) (t,u)−hψ (k) (t,u), x i
k
is the Laplace transform of a sub-stochastic measure on K. Together with
(k)
the properties (2.3)–(2.4) we may thus conclude that Qt induces a Feller
semigroup on C0 (K), which is affine in y = x/k. Hence, for each k ≥ 1, we
have constructed a k th root of X, which is by the definition of its characteristic exponents φ(k) , ψ (k) stochastically continuous. Whence Theorem 3.6.6 is
proved.
Remark 3.6.7. A consequence of the above theorem is that any affine process
on R+ or on the two-dimensional Lorentz cone is infinitely divisible. This
also follows from Duffie et al. [2003, Theorem 2.15]. For all other irreducible
symmetric cones of dimension greater than 2, the boundary is curved implying
the condition b d(r − 1)α such that the marginal distributions of an affine
process can only be infinitely divisible if α = 0.
110
3.7
Chapter 3. Affine Processes on Symmetric Cones
Results for Positive Semidefinite Matrices
For reasons of practical relevance and potential applications in mathematical
finance, we reformulate and summarize in the following theorem the previous results in the specific context of positive semidefinite matrices. These
results are also published in Cuchiero et al. [2011a] and were slightly refined
in Mayerhofer [2011].
Theorem 3.7.1. Let X be an affine process on Sr+ with r > 1. Then X is
regular and has the Feller property. Moreover, φ and ψ satisfy the generalized
Riccati equations for u ∈ Sr+ , that is,
∂φ(t, u)
= F (ψ(t, u)),
∂t
∂ψ(t, u)
= R(ψ(t, u)),
∂t
φ(0, u) = 0,
(3.70)
ψ(0, u) = u,
(3.71)
and there exists an admissible parameter set (α, b, B, c, γ, m, µ) associated with
the truncation function χ = 0 such that the functions F and R are of the
following form
Z
(e−hu,ξi − 1)m(dξ),
Z
>
e−hu,ξi − 1 µ(dξ).
R(u) = −2uαu + B (u) + γ −
F (u) = hb, ui + c −
Sr+
Sr+
Conversely, let (α, b, B, c, γ, m, µ) be an admissible parameter set. Then
there exists a unique affine process on Sr+ such that (2.1) holds for all (t, u) ∈
R+ × Sr+ , where φ(t, u) and ψ(t, u) are given by (3.70) and (3.71).
Proof. The first assertion is just a reformulation of Theorem 3.3.3 and an
application of Proposition 3.2.3. The second one follows from Theorem 3.4.13.
The following theorem suggests that affine processes on Sr+ can be seen as
solutions of some (generalized) stochastic differential equations with jumps.
Theorem 3.7.2. Let X be a conservative affine process on Sr+ with r > 1
and let (α, b, B, c = 0, γ = 0, m, µ) be the related admissible parameter set
associated with the truncation function χ = 0. Then X is a semimartingale,
3.7. Results for Positive Semidefinite Matrices
111
whose characteristics (B, C, ν) with respect to χ = 0 are given by
Z t
Cijkl (Xs )ds,
Ct,ijkl =
0
Z t
(b + B(Xs )) ds,
Bt =
(3.72)
(3.73)
0
ν(dt, dξ) = (m(dξ) + M (Xt , dξ)) dt,
(3.74)
where Cijkl (x) satisfies
Cijkl (x) = xik αjl + xil αjk + xjk αil + xjl αik ,
(3.75)
and M (x, dξ) is defined in (3.29). Furthermore, there exists, possibly on an
enlargement of the probability space, a r × r-matrix of standard Brownian
motions W such that X admits the following representation
Z t p
p Z tZ
>
Xt = x + Bt +
Xs dWs Σ + Σ dWs Xs +
ξ µX (ds, dξ),
0
0
Sr+
(3.76)
where Σ is an r × r matrix such that Σ> Σ = α and µX denotes the random
measure associated with the jumps of X.
Moreover, let X 0 be a solution of (3.76) defined on some filtered probability
space (Ω0 , F 0 , (Ft0 ), P0 ) with P0 [X0 = x]. Then P0 ◦ X 0−1 = Px .
Remark 3.7.3. By the above theorems it is easily seen that the so-called
Wishart processes, which are the unique solutions (in law) of the following
stochastic differential equation
p
p
dXt = (b + M Xt + Xt M > )dt + Xt dWt Σ + Σ> dWt> Xt ,
are particular affine processes on the cone of positive semidefinite matrices.
Proof. It follows from Theorem 1.4.8 that X is a semimartingale with characteristics (3.72)–(3.74). The canonical semimartingale representation (see Jacod and Shiryaev [2003, Theorem II.2.34]) of X is thus given by
Z tZ
c
Xt = x + Bt + Xt +
ξµX (ds, dξ),
0
Sr+
where X c denotes the continuous martingale part and µX the random measure
associated with the jumps of X. In order to establish representation (3.76),
2
we find it convenient to consider the vectorization, vec(X c ) ∈ Rr , of X c .
112
Chapter 3. Affine Processes on Symmetric Cones
f on a possibly
The aim is now to find an r2 -dimensional Brownian motion W
2
2
enlarged probability space and an r × r -matrix-valued function σ such that
Z t
c
fs .
vec(Xt ) =
σ(Xs )dW
(3.77)
0
Thus σ has to fulfill
c
it = Xt,ik αjl + Xt,il αjk + Xt,jk αil + Xt,jl αik = (σ(Xt )σ > (Xt ))ijkl .
dhXijc , Xkl
(3.78)
√
√
Defining the entries of the r2 ×r2 -matrix σ(x) by σijkl (x) = xik Σlj +Σ>
il xjk ,
yields Cijkl (x) = (σ(x)σ > (x))ijkl . Hence σ(x) satisfies (3.78). Analogous to
the proof of Rogers and Williams [1987, Theorem V.20.1], we can now build
f on an enlargement of the probability
an r2 -dimensional Brownian motion W
space such that (3.77) holds true. As the (ij)th entry of X c is given by
c
Xt,ij
=
vec(Xtc )ij
Z
=
t
d
X
fs,kl
σijkl (Xs )dW
0 k,l=1
Z t p
p >
>
=
Xs dWs Σ + Σ dWs Xs ,
ij
0
f , we
where W is the r × r-matrix Brownian motion satisfying vec(W ) = W
obtain the desired representation.
Concerning the second part of the theorem, let A denote the integrodifferential operator defined on Cb2 (Sr+ ) by
Af (x) =
X
∂ 2 f (x)
∂f (x)
1 X
Cijkl (x)
+
(bij + Bij (x))
2 i,j,k,l
∂xij ∂xkl
∂xij
i,j
Z
(f (x + ξ) − f (x)) (m(dξ) + M (x, dξ)).
+
Sr+
Then our assumption implies that P0 is a solution of the martingale problem
for A, meaning that
Z t
0
0
0
f (Xt ) − f (X0 ) −
Af (Xs−
)ds
0
is a martingale for all f ∈ Cc2 (Sr+ ). Since Px is the unique solution of the
martingale problem on (Ω, F) (see Cuchiero et al. [2011a, Proof of Proposition
5.9]), we thus have P0 ◦ X 0−1 = Px .
Part II
Polynomial Processes
113
Chapter 4
Characterization and Relation
to Semimartingales
Similar to affine processes, we define polynomial processes as a particular class
of time-homogeneous Markov processes with state space S ⊆ Rn (augmented
by a point ∆ ∈
/ S), where S denotes some closed subset of Rn . We use the
same setting as in Chapter 1, with the only difference that we assume a priori
that the time-homogeneous (not necessarily conservative) Markov processes
X defined on a filtered space (Ω, F, (Ft )t≥0 ) has càdlàg paths and that the
filtration (Ft ) is right-continuous.
Recall that we considered affine processes on the space (Ω, F, (Ft )t≥0 , Px ),
where Ω is the space of càdlàg paths defined in Remark 1.2.12 and F, Ft
are given in (1.42). Since (Ω, F, (Ft )t≥0 , Px ) does not satisfy the usual conditions, we here do not assume them either and require the filtration only
to be right-continuous such that affine processes under moment conditions
can be regarded as a subclass of polynomial processes (see Definition 4.1.1
below). This is in line with the setting of Jacod and Shiryaev [2003], where
the stochastic basis is not assumed to be complete either. As the selection of
versions with more regular trajectories might be a delicate issue, we always
assume that semimartingales have càdlàg trajectories.
Moreover, in contrast to Chapter 1, we do not restrict the definition of the
Markov property to bounded functions, but rather assume that
Ex [f (Xt+s )|Fs ] = EXs [f (Xt )],
Px -a.s.
holds for all x ∈ S∆ , s, t ∈ [0, ∞) and all Borel functions f : S∆ → R satisfying
Ex [|f (Xt )|] < ∞ for all t ≥ 0 and x ∈ S. Similarly, the semigroup (Pt )t≥0 ,
Z
Pt f (x) := Ex [f (Xt )] =
f (ξ)pt (x, dξ), x ∈ S∆ ,
S
115
116
Chapter 4. Characterization and Relation to Semimartingales
is also defined for all Borel measurable functions f : S∆ → R satisfying
Ex [|f (Xt )|] < ∞ for all t ≥ 0 and x ∈ S.
4.1
Definition and Characterization
We denote by Pol≤m (S) the finite dimensional vector space of polynomials up
to degree m ∈ N on S, i.e., the restriction of polynomials on Rn to S, defined
by


m


X
k
Pol≤m (S) := S 3 x 7→
αk x , ∆ 7→ 0 αk ∈ R ,


|k|=0
where we use multi-index notation k = (k1 , . . . , kn ) ∈ Nn , |k| = k1 + · · · + kn
and xk = xk11 · · · xknn . The dimension of Pol≤m (S) is denoted by N < ∞ and
depends on the state space S.
Definition 4.1.1. We call an S∆ -valued time-homogeneous Markov process
m-polynomial if
(i) for all 0 ≤ k ≤ m, all f ∈ Pol≤k (S), x ∈ S and t ≥ 0,
x 7→ Pt f (x) ∈ Pol≤k (S),
(ii) t 7→ Pt f (x) is continuous at t = 0 for all f ∈ Pol≤m (S).
If X is m-polynomial for all m ≥ 0, then it is called polynomial.
Remark 4.1.2.
(i) From the above definition it follows that
Pt |f |(x) = Ex [|f (Xt )|] < ∞
for every f ∈ Pol≤m (S), x ∈ S and t ≥ 0. In other words, an mpolynomial process always satisfies Ex [kXt km ] < ∞ for all x ∈ S and
t ≥ 0.
(ii) The subtlety of Definition 4.1.1 lies in the fact that we assume
Pt Pol≤k (S) ⊂ Pol≤k (S)
for all 0 ≤ k ≤ m (compare with Remark 4.1.10 (iv)). The assumption
that Pt Pol≤m (S) ⊂ Pol≤m (S) only for m, but not for smaller degrees is
not sufficient for our proof of the equivalence of (iv) in Theorem 4.1.8
to the other assertions. The equivalence of (i), (ii), (iii) can however be
proved by the same arguments.
4.1. Definition and Characterization
117
(iii) Throughout this chapter we shall always use continuity assumptions of
the type “ t 7→ Pt f (x) is continuous for every x ∈ S ”, which are easier
to verify than strong continuity with respect to some chosen norm. For
f ∈ Pol≤m (S), the existence of moments of degree m + ε for some ε > 0
is for examples already sufficient.
Let us introduce the following two notions of Markov generators, which we
shall use to characterize m-polynomial processes.
Definition 4.1.3. An operator A with domain DA is called infinitesimal generator for X if DA consists of those functions f : S∆ → R which satisfy
Pt |f |(x) < ∞ for all t ≥ 0 and x ∈ S and for which there exists a function
Af such that the process
Z t
Af (Xs )ds,
(4.1)
f (Xt ) − f (X0 ) −
0
is a (Ft , Px )-martingale for every x ∈ S∆ .
Remark 4.1.4. If f lies in the domain of the infinitesimal generator,
then
Rt
due to the martingale property all increments of f (Xt ) − f (X0 ) − 0 Af (Xs )ds
have vanishing expectation, i.e., for all s < t,
Z t
Ex f (Xt ) − f (Xs ) −
Af (Xu )du = 0.
s
Rt
In particular, by Fubini’s theorem, 0 Pu Af (x)du exists on finite time intervals
and thus also Pu |Af |(x) for almost all u with respect to the Lebesgue measure.
The following notion of the extended infinitesimal generator is due to
Dynkin (see, e.g., Çinlar et al. [1980, Definition 7.1]).
Definition 4.1.5. An operator G with domain DG is called extended infinitesimal generator for X if DG consists of those Borel measurable functions
f : S∆ → R for which there exists a function Gf such that the process
Z t
f (Xt ) − f (X0 ) −
Gf (Xs )ds
(4.2)
0
is well-defined and a (Ft , Px )-local martingale for every x ∈ S∆ .
Remark 4.1.6. As in Chapter 1, we define the lifetime of the process by
T∆ (ω) = inf{t | Xt (ω) = ∆},
(4.3)
118
Chapter 4. Characterization and Relation to Semimartingales
where the infimum over the empty set is set to be ∞. Since {T∆ < t} =
S
q<t,q∈Q {Xq = ∆} ∈ Ft and as (Ft ) is supposed to be right-continuous, T∆ is
an Ft -stopping time. Due to our convention f (∆) = 0, the local martingale
property of (4.2) is therefore equivalent to
Z t
Gf (Xs )1{s<T∆ } ds
f (Xt )1{t<T∆ } − f (X0 ) −
0
being a local martingale. The analogous statement holds true for (4.1).
Since our definition of the infinitesimal generator in (4.1) is not the standard one used in the literature for Feller processes (compare Revuz and Yor
[1999, Chapter VII]), we formulate the following lemma.
Lemma 4.1.7. If f lies in the domain of the infinitesimal generator, f ∈ DA ,
then we have:
(i) Pt f ∈ DA and APt f = Pt Af for every t ≥ 0.
(ii) If t 7→ Pt Af (x) is continuous at t = 0, then Pt f is solves the Kolmogorov backward equation in the strong sense, i.e.,
∂u(t, x)
= Au(t, x),
∂t
u(0, x) = f (x).
Proof. For the first statement, we show that
Z h
Pt f (Xh ) − Pt f (X0 ) −
Pt Af (Xs )ds
0
is a (Fh , Px )-martingale for any fixed t ≥ 0. By the definition of the infinitesimal generator, this then implies that Pt f ∈ DA and APt f = Pt Af . Indeed,
since f lies in DA , we have by Definition 4.1.3 and Remark 4.1.4 that f (Xs )
and Af (Xs ) are integrable for every s ≥ 0, hence Pt f (Xs ) and Pt Af (Xs ) as
well. Therefore the following expectation is well-defined and we obtain for
u≤h
Z h
Ex Pt f (Xh ) − Pt f (X0 ) −
Pt Af (Xs )ds Fu
Z 0u
= Pt f (Xu ) − Pt f (x) −
Pt Af (Xs )ds
0
Z h
+ Ex Pt f (Xh ) − Pt f (Xu ) −
Pt Af (Xs )ds Fu .
u
4.1. Definition and Characterization
119
By the Markov property, the conditional expectation on the right is equal to
Z h−u
EXu Pt f (Xh−u ) − Pt f (X0 ) −
Pt Af (Xs )ds .
0
But for any y ∈ S∆ , we have
Z h−u
Pt Af (Xs )ds
Ey Pt f (Xh−u ) − Pt f (X0 ) −
0
Z t+h−u
Ps̃ Af (y)ds̃
= Pt+h−u f (y) − Pt f (y) −
t
= 0,
where the last equality follows from Remark 4.1.4. This completes the proof
of (i).
Statement (ii) follows from the continuity of t 7→ Pt Af (x) and from assertion (i), since
∂Pt f (x)
Ph f (x) − f (x)
Pt+h f (x) − Pt f (x)
= lim
= lim Pt
h→0
h→0
∂t
h
h
Z h
1
Ps Af (x)ds = Pt Af (x) = APt f (x).
= lim Pt
h→0
h 0
Let us now state our main theorem:
Theorem 4.1.8. Let X be a time-homogeneous Markov process with state
space S∆ and semigroup (Pt ). Additionally we assume that t 7→ Pt f (x) is
well-defined and continuous at t = 0 for all f ∈ Pol≤m (S). Then the following
assertions are equivalent:
(i) X is m-polynomial for some m ≥ 0.
(ii) For every 0 ≤ k ≤ m, there exists a linear map A : Pol≤k (S) →
Pol≤k (S), such that, for all t ≥ 0, (Pt ) restricted to Pol≤k (S) can be
written as
Pt |Pol≤k (S) = etA .
(iii) Pol≤m (S) lies in the domain of the infinitesimal generator, i.e., for all
f ∈ Pol≤m (S), x ∈ S∆ and t ≥ 0,
Z t
f (Xt ) − f (X0 ) −
Af (Xs )ds
0
is a (Ft , Px )-martingale, and A(Pol≤k (S)) ⊂ Pol≤k (S) for all 0 ≤ k ≤
m.
120
Chapter 4. Characterization and Relation to Semimartingales
Moreover, the following weaker statement is obviously implied by (i), (ii) and
(iii), but is also equivalent if m ≥ 2 is an even number.
(iv) Pol≤m (S) lies in the domain of the extended infinitesimal generator, i.e.,
for all f ∈ Pol≤m (S), x ∈ S∆ and t ≥ 0,
Mtf
Z
t
Gf (Xs )ds
:= f (Xt ) − f (X0 ) −
0
is a (Ft , Px )-local martingale and G(Pol≤k (S)) ⊂ Pol≤k (S) for all 0 ≤
k ≤ m.
Remark 4.1.9. From the point of view of applications the most important
implication is (iv) ⇒ (ii) (see Chapter 6).
Proof. Our strategy to prove the above equivalences is to show (i) ⇒ (ii) ⇒
(iii) ⇒ (i) and (iv) ⇒ (iii) under the additional assumption that m ≥ 2 is
an even number. In order to prove the last implication, we need some results
derived in Proposition 4.2.1 and Lemma 4.2.3 below. Therefore we postpone
the proof of (iv) ⇒ (iii) to Subsection 4.3.1.
Throughout the proof, let 0 ≤ k ≤ m be fixed. We start by showing (i)
⇒ (ii). Let L(Pol≤k (S)) denote the space of all linear maps from Pol≤k (S) to
Pol≤k (S). By the semigroup property,
P(·) |Pol≤k (S) : R+ → L(Pol≤k (S))
(4.4)
satisfies the Cauchy functional equation
Pt+s = Pt Ps for all t, s ≥ 0,
P0 = I.
Since Pol≤k (S) is finite dimensional, continuity of (4.4) at t = 0 already implies
that there exists some A ∈ L(Pol≤k (S)) such that Pt |Pol≤k (S) = etA (see Engel
and Nagel [2000, Theorem I.2.9]).
Next, we show (ii) ⇒ (iii). By (ii) we have, for every f ∈ Pol≤k (S),
Af ∈ Pol≤k (S) and
Z
Pt f − f −
t
tA
Z
Ps Af ds = e f − f −
0
t
esA Af ds = 0.
0
Rt
Thus f (Xt ) − f (x) − 0 Af (Xs )ds is a (Ft , Px )-martingale. Hence f lies in DA
and Af = Af , implying that A(Pol≤k (S)) ⊂ Pol≤k (S) holds true.
4.1. Definition and Characterization
121
In order to prove (iii) ⇒ (i), we consider the Kolmogorov backward equation (in the strong sense) for an initial value u(0, ·) = f ∈ Pol≤k (S):
∂u(t, x)
= Au(t, x).
∂t
By Lemma 4.1.7 (ii), Pt f solves the Kolmogorov equation (in the strong sense),
since t 7→ Pt Af (x) is continuous at t = 0 for any f ∈ Pol≤k (S). This follows
from the fact that A maps Pol≤k (S) to itself and the continuity assumption
on t 7→ Pt f (x) for any f ∈ Pol≤k (S). By choosing a basis he1 , . . . , eN i of
Pol≤k (S), we can define a linear map A on Pol≤k (S) by setting
Aek =:
N
X
Akl el .
l=1
Then A|Pol≤k (S) = A and the Kolmogorov backward equation can be understood as a linear ODE in the classical sense, whose unique solution is given
by etA f . The map Ps (e(t−s)A f ) is therefore constant with respect to s, since
its first derivative vanishes, that is,
d
Ps e(t−s)A f (x) = Ps Ae(t−s)A f (x) − Ps Ae(t−s)A f (x) = 0,
ds
where we apply Lemma 4.1.7 (i). Hence, on Pol≤k (S), Pt f is equal to etA f
and is therefore a polynomial of degree smaller than or equal to k. Since this
holds true for any 0 ≤ k ≤ m, X is m-polynomial.
Remark 4.1.10. (i) There is no need in Theorem 4.1.8 (ii) to restrict the
time parameter t to R+ , since, for t ∈ R, (etA ) extends to a group.
(ii) If X is an m-polynomial process, then the process Z = (X, X 2 , . . . , X m )
is a 1-polynomial process. If m is even, the analysis of m-polynomial
processes could be reduced to the study of 2-polynomial processes at the
cost of a more complicated state space, which is due to the construction
m
Z 0 = (X, X 2 , . . . , X 2 ).
(iii) Let us remark that the implication (iv) ⇒ (iii) in Theorem 4.1.8 only
holds true if m ≥ 2. In the case m = 1, the inverse 3-dimensional
Bessel process X is an example showing that the extended infinitesimal
generator maps Pol≤1 (R+ ) to Pol0 (R+ ), while
Z t
Xt − X0 −
GXs ds = Xt − X0
(4.5)
0
122
Chapter 4. Characterization and Relation to Semimartingales
is a strict local martingale (see Protter [2005, Example I.6.2]). Indeed,
1
the inverse 3-dimensional Bessel process defined by X = kBk
, where B
denotes a 3-dimensional Brownian motion started at B0 6= 0, satisfies
dXt = −Xt2 dWt ,
X0 =
1
,
kB0 k
where W is a one-dimensional standard Brownian motion. The extended
infinitesimal generator is therefore given by
1 d2 f (x)
Gf (x) = x4
.
2
dx2
Hence G(Pol≤1 (R+ )) = 0. However, since (4.5) fails to be a true martingale, X is not a 1-polynomial process.
(iv) In Definition 4.1.1 we require m-polynomial processes to be also k-polynomial for all 0 ≤ k ≤ m, that is, we implicitly exclude processes whose
extended infinitesimal generator maps polynomials of degree k < m
to polynomials of degree greater than k ≤ m, while G(Pol≤m (S)) ⊂
Pol≤m (S) still holds true. The following process is an example of this
type. Consider
q
1 2
1
− bXt + Xt dt + Xt2 (1 − Xt )dWt , X0 = x ∈ [0, 1],
dXt =
2
2
where b ≥ 1 and W is a one-dimensional standard Brownian motion.
In this case, the state space is the interval S = [0, 1]. Existence of weak
solutions for this SDE follow from the continuity of the drift and the
diffusion component (see Rogers and Williams [1987, Theorem V.23.5]).
Moreover, we have
1 2 df (x) 1 2
d2 f (x)
1
− bx + x
+ x (1 − x)
.
Gf (x) =
2
2
dx
2
dx2
Thus G(Pol≤1 (S)) ⊂ Pol≤2 (S), while G(Pol≤2 (S)) ⊂ Pol≤2 (S). Using
the arguments of Lemma 4.2.3 below, it follows that
Z t
f (Xt ) − f (X0 ) −
Gf (Xs )ds
0
is a true martingale for f ∈ Pol≤2 (S) due to compactness of the state
space. By the same arguments as in the proof of Theorem 4.1.8 (iii)
⇒ (i), it then follows that Pt (Pol≤2 (S)) ⊂ Pol≤2 (S) but Pt (Pol≤1 (S)) *
Pol≤1 (S).
4.2. Polynomial Semimartingales
123
The following corollary is a basic conclusion of Theorem 4.1.8 (ii) and
establishes a link to time-space harmonic functions (see, e.g., Solé and Utzet
[2008]).
Corollary 4.1.11. Let X be an m-polynomial process with semigroup (Pt ) and
let f ∈ Pol≤m (S) be fixed. Then there exists a unique function Q : R×S∆ → R,
being real analytic in time and satisfying Q(t, ·) ∈ Pol≤m (S) for all t ∈ R such
that Q(0, x) = f (x) and Q(t − s, Xs ) is a (Fs , Px )-martingale for s ≥ 0.
Proof. We prove that Q(t, x) is given by Pt f (x). As X is m-polynomial,
Pt f ∈ Pol≤m (S) and by Theorem 4.1.8 (ii) and Remark 4.1.10 (i) it is a
real analytic function in time. Clearly, P0 f (x) = f (x) and by the Markov
property Pt−s f (Xs ) is a (Fs , Px )-martingale. Concerning uniqueness, let now
Q : R × S∆ → R be a function satisfying Q(0, x) = f (x) and the martingale
property. The latter implies for all s ≥ 0
Q(t, x) = Ex [Q(t − s, Xs )].
Thus, by setting s = t, we obtain
Q(t, x) = Ex [Q(0, Xt )] = Ex [f (Xt )] = Pt f (x),
as Q(0, Xt ) = f (Xt ). This proves uniqueness.
Remark 4.1.12. For t = 0, Q(−s, x) as defined in Corollary 4.1.11 can be
considered as a time-space harmonic function (see, e.g., Solé and Utzet [2008])
for the m-polynomial process X.
4.2
Polynomial Semimartingales
In our setting defined above we neither assumed the Markov process to be conservative, nor specified the domain of the (extended) infinitesimal generator,
for example by requiring the family of functions {eihu,·i | u ∈ Rn } to lie in DA or
DG . Therefore the process X with X0 = x is in general not a semimartingale
with respect to the stochastic basis (Ω, F, (Ft ), Px ). However, if a Markov
process satisfies the assumption of Theorem 4.1.8 (iv), then (Xt 1{t<T∆ } ) is
a special semimartingale whose characteristics are absolutely continuous with
respect to the Lebesgue measure, as shown in Proposition 4.2.1 below. This remarkable property follows from the assumption that G(Pol≤m (S)) ⊂ Pol≤m (S)
and compensates for the fact that Pol≤m (S) is not a complete class in the terminology of Çinlar et al. [1980, Definition 7.8]. Indeed, according to Çinlar
et al. [1980, Theorem 7.16] a Markov process is a semimartingale with absolutely continuous characteristics if and only if the domain of its extended
generator is a full and complete class.
124
Chapter 4. Characterization and Relation to Semimartingales
Proposition 4.2.1. Let X be a time-homogeneous Markov process with state
space S∆ . Let m ≥ 2 and assume that Pol≤m (S) lies in the domain of the
extended infinitesimal generator and that G maps Pol≤k (S) to itself for all
0 ≤ k ≤ m. Then (Xt 1{t<T∆ } ) is a special semimartingale with respect to
the stochastic basis (Ω, F, (Ft ), Px ). The components of its characteristics
(B, C, ν) associated with the “truncation function” χ(ξ) = ξ satisfy1
t
Z
t
Z
bi (Xs )1{s<T∆ } ds,
(GXs,i )1{s<T∆ } ds =:
0
0
Z t
Z tZ
aij (Xs )1{s<T∆ } ds,
ξi ξj ν(ds, dξ) =:
+
(4.6)
Bt,i =
Ct,ij
0
Rn
(4.7)
0
where bi ∈ Pol≤1 (S) and aij ∈ Pol≤2 (S). Moreover, the characteristics C and
ν can be written as
Z t
cs,ij ds, ν(ω; dt, dξ) = Kω,t (dξ)dt,
(4.8)
Ct,ij =
0
where (cij )i,j≤n is a predictable process and Kω,t (dξ) is a predictable random
measure on (Rn , B(Rn )). Finally, we have for all 3 ≤ |k| ≤ m
Z
k
ξ Kω,t (dξ) =
Rn
|k|
X
αl Xtl (ω)1{t<T∆ (ω)} ,
for almost all t ≥ 0,
(4.9)
|l|=0
with some coefficients αl and for all 2 ≤ k ≤ 2b m2 c
Z
c
e 1 + kXt (ω)1{t<T (ω)} k2b k+1
2
kξkk Kω,t (dξ) ≤ C
,
∆
(4.10)
Rn
e
for almost all t ≥ 0 and some constant C.
Remark 4.2.2. (i) Due to the fact that Gf (x)Ris a well-defined polynomial
for every f ∈ Pol≤m (S), it follows that RnR ξ k Kω,t (dξ) exists for all
2 ≤ |k| ≤ m. This means in particular that Rn kξkk Kω,t (dξ) < ∞ for
all 2 ≤ k ≤ m.
(ii) We shall make use of the very condensed notation GXs,i := G pri (Xs ),
where we consider the projection on the i-th component. Analogously we
apply this to quadratic monomials.
1
All statements concerning the characteristics are meant up to an evanescent set.
4.2. Polynomial Semimartingales
125
(iii) The above proposition asserts that a Markov process which satisfies the
assumption of Theorem 4.1.8 (iv) is a semimartingale with absolutely
continuous characteristics. In particular, the drift part is an affine function in X, the modified second characteristic, which corresponds to
Z tZ
ξi ξj ν(ds, dξ)
Ct,ij +
0
Rn
(see Jacod and Shiryaev [2003, Definition II.2.16, Proposition II.2.17]),
is a quadratic polynomial, and the moments of the compensator of the
jump measure are polynomials in X of the same degree.
Proof. For all f ∈ Pol≤m (S)
Mtf
t
Z
= f (Xt )1{t<T∆ } − f (X0 ) −
(Gf (Xs ))1{s<T∆ } ds,
(4.11)
0
is a (Ft , Px )-local
R t martingale. Here, T∆ denotes the lifetime defined in (4.3).
As the process 0 (Gf (Xs ))1{s<T∆ } ds is predictable, (f (Xt )1{t<T∆ } ) is a special
R-valued semimartingale for all f ∈ Pol≤m (S). In particular, f (Xt )1{t<T∆ }
has càdlàg paths implying that |f (XT∆ − )| < ∞ (notice here that f (∆) =
0) and f (X) cannot explode. Choosing f (x) = xi for i = 1, . . . , n, then
implies that (Xt 1{t<T∆ } ) is an (n-dimensional) special semimartingale. Let
now (B, C, ν) denote the characteristics of (Xt 1{t<T∆ } ) with respect to the
“truncation function” χ(ξ) = ξ. Since
Xt,i 1{t<T∆ } = X0,i +
Mtxi
Z
t
(GXs,i )1{s<T∆ } ds,
+
0
it follows by the unique decomposition of a special semimartingale into a local
martingale and a predictable finite variation process that
Z
Bt =
t
(GXs,i )1{s<T∆ } ds.
0
Setting bi (x) := Gxi (notice here Remark 4.2.2 (ii)) implies that bi ∈ Pol≤1 (S).
In order to determine the properties of C and ν, let us consider the polynomials f (x) = xi xj for 1 ≤ i, j ≤ n. Then the special semimartingale
Xt,i Xt,j 1{t<T∆ } can be written as
Xt,i Xt,j 1{t<T∆ } = X0,i X0,j +
xx
Mt i j
Z
+
t
(GXs,i Xs,j )1{s<T∆ } ds.
0
(4.12)
126
Chapter 4. Characterization and Relation to Semimartingales
By Itô’s formula, we have on the other hand
Z t
Xs−,i 1{s<T∆ } dMsxj
Xt,i Xt,j 1{t<T∆ } = X0,i X0,j +
0
Z t
Z t
xi
+
Xs−,j 1{s<T∆ } dMs +
Xs,i 1{s<T∆ } bj (Xs )ds
0
0
(4.13)
Z t
+
Xs,j 1{s<T∆ } bi (Xs )ds + Ct,ij
0
Z tZ
+
ξi ξj µX1{t<T∆ } (ds, dξ),
0
Rn
where µX1{t<T∆ } denotes the random measure associated with the jumps of
X1{t<T∆ } . Since M xi is a local martingale and Xs−,j is càglàd,
Z t
Xs−,j 1{s<T∆ } dMsxi
0
is also a local martingale (see, e.g., Protter [2005, Theorem III.7.33]) for all
i, j = 1, . . . , n. Furthermore,
Z t
Z t
Xs,i 1{s<T∆ } bj (Xs )ds +
Xs,j 1{s<T∆ } bi (Xs )ds + Ct,ij
0
0
is a predictable process of finite variation, thus in particular a process of
locally integrable variation by Jacod and Shiryaev [2003, Lemma I.3.10]. As
Xt,i Xt,j 1{t<T∆ } is a special semimartingale, it follows from Jacod and Shiryaev
[2003, Proposition I.4.22] that
Z tZ
ξi ξj µX1{t<T∆ } (ds, dξ)
0
Rn
is also of locally integrable variation, since it is of finite variation and for a
special semimartingale the finite variation part is locally integrable. Therefore
we have by Jacod and Shiryaev [2003, Proposition II.1.28] that
Z tZ
Z tZ
X1{t<T∆ }
ξi ξj µ
(ds, dξ) −
ξi ξj ν(ds, dξ)
0
Rn
0
Rn
is a local martingale. Thus, putting (4.12) and (4.13) together, we find
Z t
Z t
(GXs,i Xs,j )1{s<T∆ } ds =
Xs,i bj (Xs )1{s<T∆ } ds
0
0
Z t
+
Xs,j bi (Xs )1{s<T∆ } ds + Ct,ij
0
Z tZ
+
ξi ξj ν(ds, dξ).
0
Rn
4.2. Polynomial Semimartingales
127
Since Gxi xj and xi bj (x) lie in Pol≤2 (S), we have
Z tZ
Ct,ij +
Z
t
aij (Xs )1{s<T∆ } ds
ξi ξj ν(ds, dξ) =
0
Rn
(4.14)
0
for some aij ∈ Pol≤2 (S).
RtR
Let us now define A0t (ω) = 0 Rn kξk2 ν(ω; ds, dξ). By the same arguments
as in Jacod and Shiryaev [2003, Proposition II.2.9.b], there exists a random
measure K 0 (ω, t; dξ) on (Rn , B(Rn )) such that ν(ω; dt, dξ) = K 0 (ω, t; dξ)dAt (ω).
Moreover, since
n
X
Ct,ii (ω) +
A0t (ω)
i=1
=
n Z
X
i=1
t
Z
t
aii (Xs (ω))1{s<T∆ (ω)} ds =:
0
as (ω)ds
0
and as Ct,ii , i = 1, . . . , n, and A0t are nonnegative increasing processes (of finite
variation), Cii and A0 are absolutely continuous with respect to the Lebesgue
measure. Hence Jacod and Shiryaev [2003, Proposition I.3.13]
implies the
Rt
existence of predictable processes cii and H such that Ct,ii = 0 cs,ii ds and A0t =
Rt
Hs ds. Then Kω,t (dξ) = Ht (ω)K 0 (ω, t; dξ) is again a predictable random
0
measure satisfying ν(ω; dt, dξ) = Kω,t (dξ)ds Px -a.s. Having constructed this
kernel, (4.14) now becomes
Ct,ij
Z t
Z
=
aij (Xs )1{s<T∆ } −
0
ξi ξj Kω,t (dξ) ds,
Rn
implying that Ct,ij , for i 6= j, is also absolutely continuous with respect to the
Rt
Lebesgue measure and can therefore be written as Ct,ij = 0 cs,ij ds.
In order to establish properties (4.9) and (4.10), we consider the polynomial
128
Chapter 4. Characterization and Relation to Semimartingales
f (x) = xk for 3 ≤ k = |k| ≤ m. As before, we obtain by Itô’s formula
f (Xt )1{t<T∆ } − f (X0 )
n Z t
n Z t
X
X
xi
Di f (Xs )1{s<T∆ } bi (Xs )ds
Di f (Xs− )1{s<T∆ } dMs +
=
0
i=1
n
X
i=1
1
+
2 i,j=1
Z t
Z tZ

+
0
Rn
+
0
Rn
0
k X
k

|l|=3
l
k X
k

|l|=2
l

Xsk−l 1{s<T∆ } ξ l  Kω,s (dξ)ds

k−l
Xs−
1{s<T∆ } ξ l  µX1{t<T∆ } (ds, dξ)

k X
k

Xsk−l 1{s<T∆ } ξ l  Kω,s (dξ)ds.
l
Rn
Z tZ
−
ξi ξj Kω,s (dξ)
ds
Rn
0

Z tZ
0
Z
Dij f (Xs )1{s<T∆ } cs,ij +

|l|=2
(4.15)
Since f (Xt )1{t<T∆ } is a special semimartingale, it follows by the same arguments as before that


Z tZ
k X
k
k−l

Xs−
1{s<T∆ } ξ l  µX1{t<T∆ } (ds, dξ)
l
0
Rn
|l|=2
is of locally integrable variation. This implies that the difference of the last
two terms in (4.15)
is a local martingale. As Gf (x) = Gxk , Di f (x)b(x) and
R
Dij f (x)(cij + Rn ξi ξj Kω,· (dξ)) lie in Pol≤k (S) for all k = |k| ≤ m, (4.9) then
follows from (4.11) and induction. Moreover, since 2b k+1
c is even, we have
2
m
for 2 ≤ k ≤ 2b 2 c and almost all t ≥ 0
Z
Z
Z
k+1
k
2
kξk Kω,t (dξ) ≤
kξk Kω,t (dξ) +
kξk2b 2 c Kω,t (dξ)
Rn
Rn
≤
n
X
Rn
c
2b k+1
2
aii (Xt )1{t<T∆ } +
i=1
αl Xtl 1{t<T∆ }
|l|=0
2b k+1
c
2
e 1 + kXt 1{t<T } k
≤C
∆
e denotes some constant.
where C
X
,
4.2. Polynomial Semimartingales
129
Using the above derived semimartingale properties, we can now establish
a maximal inequality, which we need for the proof of the implication (iv) ⇒
(iii) in Theorem 4.1.8. A similar statement for the case of Lévy driven SDEs
can be found in Jacod, Kurtz, Méléard, and Protter [2005].
Lemma 4.2.3. Fix T > 0 and let m ≥ 2. Let (Xt 1{t<T∆ } ) be a special
semimartingale with state space S, whose characteristics (B, C, ν) associated
with the “truncation function” χ(ξ) = ξ satisfy (4.6), (4.7) and (4.8). Then
e such that for all 0 ≤ t ≤ T
there exists a constant C
Z t
m
m
e
Ex kXs 1{s<T∆ } km ds
≤ C kxk + 1 +
Ex sup Xs 1{s<T∆ }
s≤t
0
Z
+
t
Z
Ex
0
!
(4.16)
kξkm Kω,s (dξ) ds .
Rn
Remark 4.2.4. Notice that this lemma remains also true when one of both
sides is infinite. The lemma should be considered as an assertion about special
semimartigales with a particular structure on the characteristics.
Proof. Let Xt 1{t<T∆ } = X0 + Mt + Bt be the canonical decomposition of
the special semimartingale (Xt 1{t<T∆ } ). By the following version of Jensen’s
inequality
m
k
k
X
X
m−1
y
≤
k
|yi |m
(4.17)
i
i=1
i=1
for m ≥ 1, we have
m
m
m
m−1
m
sup Xs,i 1{s<T∆ } ≤ 3
|X0,i | + sup |Ms,i | + sup |Bs,i |
.
s≤t
s≤t
s≤t
Thus, in order to prove (4.16), it suffices to find estimates for sups≤t |Ms,i | and
e in the inequalities below may
sups≤t |Bs,i |. We remark that the constant
C
Rs
vary from line to line. Since Bs,i = 0 bi (Xu )1{u<T∆ } du and bi ∈ Pol≤1 (S), we
have
Z s
m Z t
m
sup bi (Xu )1{u<T∆ } du ≤
|bi (Xu )| 1{u<T∆ } du
s≤t
0
0
Z t
(4.18)
≤
tm−1 |bi (Xu )|m 1{u<T∆ } du
0
Z t
m
e
≤C 1+
Xu 1{u<T∆ } du .
0
130
Chapter 4. Characterization and Relation to Semimartingales
Hence, by Fubini’s theorem,
Z t
m
m
e 1+
Ex kXs 1{t<T∆ } k ds .
Ex sup |Bs,i | ≤ C
s≤t
0
Concerning sups≤t |Ms,i |, an application of Burkholder-Davis-Gundy’s inequality yields
h
mi
m
e x [Mi , Mi ]t2 .
Ex sup |Ms,i | ≤ CE
s≤t
Since the finite variation process Bt,i is continuous, we have
Z tZ
[Mi , Mi ]t = Ct,ii +
ξi2 µX1{t<T∆ } (ds, dξ) =: Ct,ii + Yt ,
0
Rn
where µX1{t<T∆ } is the random measure associated with the jumps of X1{t<T∆ } .
m
m
m
m
2
Applying again inequality (4.17), we obtain [Mi , Mi ]t2 ≤ 2 2 −1 (Ct,ii
+ Yt 2 ).
Rt
As C satisfies (4.7), we can estimate Ct,ii by Ct,ii ≤ 0 aii (Xs )1{s<T∆ } ds, where
aii ≥ 0 ∈ Pol≤2 (S). Using a similar reasoning as in (4.18), we get
Z t
h mi
m
2
e
Ex kXs 1{t<T∆ } k ds .
Ex Ct,ii ≤ C 1 +
0
m2
R
m
t
Therefore it remains to handle Yt 2 = 0 ξi2 µX1{t<T∆ } (ds, dξ) . Following
the approach of Jacod et al. [2005], let us define Tn = inf{t | Yt ≥ n}. Then
X
m
m
m
2
(Ys− + ∆Ys ) 2 − (Ys− ) 2
=
Yt∧T
n
s≤t∧Tn
Z
t∧Tn
Z
=
Rn
0
m
m
(Ys− + ξi2 ) 2 − (Ys− ) 2 µX1{t<T∆ } (ds, dξ),
which is due to the fact that Y is purely discontinuous, non-decreasing and
∆Ys = |∆(Xs,i 1{s<T∆ } )|2 . Furthermore, since ν is the predictable compensator
of µX1{t<T∆ } ,
Z t∧Tn Z
h m i
m
2 m
2
(Ys− + ξi ) 2 − (Ys− ) 2 ν(ds, dξ) ,
(4.19)
Ex Yt∧Tn = Ex
0
Rn
an equality which even remains true if one of both sides is infinite. In the
sequel we shall use the following inequalities (see Jacod et al. [2005])
(y + z)p − y p ≤ 2p−1 (y p−1 z + z p ),
xp
y p−1 x ≤ εy p + p−1 ,
ε
(4.20)
(4.21)
4.2. Polynomial Semimartingales
131
for x, y, z ≥ 0, every ε > 0 and p ≥ 1. Applying (4.20), equation (4.19)
becomes
Z t∧Tn Z
h m i
m
m
−1 2
m
−1
2
2
22
Ex Yt∧Tn ≤ Ex
Ys− ξi + |ξi | ν(ds, dξ) .
Rn
0
For the first part, we then have due to the assumption on ν
Z
t∧Tn
Z
2
Ex
0
m
−1
2
Rn
m
−1
2
ξi2 ν(ds, dξ)
Ys−
Z t∧Tn
= Ex
2
m
−1
2
m
−1
2
Z
Ys
0
Z
≤ Ex
Rn
t∧Tn
ξi2 Kω,s (dξ)
2
m
−1
2
m
−1
2
Ys
ds
aii (Xs )1{s<T∆ } ds ,
0
where the last inequality follows from the fact that Ys is positive for all s ≤ t.
e + kxk2 ), we obtain using (4.21),
Estimating aii (x) by C(1
Z t∧Tn h m i
m
m
1
+
kX
k
s
2
2
e εYs +
C
1{s<T∆ } ds
Ex Yt∧Tn ≤ Ex
m
ε 2 −1
0
Z t∧Tn Z
m
−1
m
2
+ Ex
2
|ξi | ν(ds, dξ)
0
Rn
h
m i
e x n m2 ∧ Y 2
≤ CεE
t∧Tn
"Z
#
t∧Tn
e
C
(1 + kXs km )1{s<T∆ } ds
+ Ex
m
−1
2
ε
0
Z t∧Tn Z
m
−1
m
2
2
+ Ex
|ξi | Kω,s (dξ)ds .
0
Rn
m
m
R t∧T
m
2
follows from the fact that Ys
The inequality 0 n Ys 2 1{s<T∆ } ds ≤ n 2 ∧ Yt∧T
n
is non-decreasing and Ys− ≤ n for s ≤ Tn . The last jump might cause that
YTn > n, but this is not felt in the integral. Choosing for example ε = 21Ce ,
leads to
h m i 1 h m
m i
1 h m2 i
2
2 ∧ Y 2
−
Ex Yt∧Tn ≤ Ex Yt∧T
E
n
x
t∧Tn
n
2
Z t∧Tn 2
Z
m
m
e
≤ CEx
1 + kXs 1{s<T∆ } k +
|ξi | Kω,s (dξ) ds ,
0
Rn
m
m
2
2
which also remains true if one of both sides is infinite. Since Yt∧T
≤ Yt∧T
,
n
n+1
132
Chapter 4. Characterization and Relation to Semimartingales
we can apply the monotone convergence theorem to obtain
h
Ex Yt
m
2
i
Z
t
Ex kXs 1{s<T∆ } km ds
e 1+
≤C
0
Z
+
t
Z
Ex
0
!
|ξi |m Kω,s (dξ) ds .
Rn
Again by (4.17) and the equivalence of the 1- and 2- norms, we finally get the
desired estimate (4.16).
4.3
Characterization by means of the Extended
Generator
We are finally prepared to prove the implication (iv) ⇒ (iii) of Theorem 4.1.8
for m ≥ 2 an even number. Based on the maximal inequality stated in
Lemma 4.2.3 above, we shall exploit
the
the assumption of
fact that
munder
Theorem 4.1.8 (iv) we have Ex sups≤t Xs 1{s<T∆ }
< ∞ whenever
m Ex Xs 1{s<T∆ } < ∞,
4.3.1
for all s ≤ t.
Proof of Theorem 4.1.8 (iv) ⇒ (iii)
Proof. Recall that M was defined by
Msf
Z
= f (Xs ) − f (X0 ) −
s
Gf (Xu )du.
0
We now show that
Ex
Z s
f
sup Ms = Ex sup f (Xs ) − f (X0 ) −
Gf (Xu )du < ∞
s≤t
s≤t
0
for every fixed t ≥ 0. The dominated convergence theorem then implies that
Mtf is a (Ft , Px )-martingale and thus Af = Gf on Pol≤m (S). Since
we have
Gf ∈ Pol≤m (S) for f ∈ Pol≤m (S), we can dominate sups≤t Msf with the
4.3. Characterization by means of the Extended Generator
133
following expression:
Z s
f
Gf (Xu )1{u<T∆ } du
sup Ms ≤ sup f (Xs )1{s<T∆ } + |f (X0 )| + sup s≤t
s≤t
s≤t
0
m
≤ C1 1 + sup Xs 1{s<T } + C2 (1 + kX0 km )
∆
s≤t
Z
+
t
m C3 1 + Xs 1{s<T∆ } ds
0
m
m
e
,
≤ C 1 + kX0 k + sup Xs 1{s<T∆ }
s≤t
e = C1 + C2 + C3 t for some constants C1 , C2 , C3 . Due to Proposiwith C
tion 4.2.1, (Xt 1{t<T∆ ) satisfies the condition of Lemma 4.2.3 above. Moreover,
by the moment assumption on X, that is, t 7→ Pt f (x) is well-defined for all
f ∈ Pol≤m (S), and the fact that m is even, Lemma 4.2.3 yields
m
Ex sup Xs 1{s<T∆ } < ∞
s≤t
for every t ≥ 0. Indeed, since m is even, this follows from (4.10), where we
have
Z
m
e 1 + Ex kXt 1{t<T } km .
kξk Kω,t (dξ) ≤ C
Ex
∆
Rn
Finally, by the above estimate, we have Ex sups≤t Msf < ∞ for every t ≥ 0,
proving the assertion.
Remark 4.3.1. If m > 2 is an odd number, the above proof implies that the
assumptions of Theorem 4.1.8 (iv) together with
Z
m
Ex
kξk Kω,t (dξ) < ∞
Rn
are sufficient for X being an m-polynomial process. In particular, the conditions of Theorem 4.1.8 (iv) are always sufficient in the case of pure diffusions
and jumps supported on the positive orthant, where the latter statement is a
consequence of (4.9) and the moment conditions on X.
4.3.2
Semimartingales which are Polynomial Processes
Corollary 4.3.2. Let X be a time-homogeneous Markov process with state
space S∆ and semigroup (Pt ) such that t 7→ Pt f (x) is continuous at t = 0
134
Chapter 4. Characterization and Relation to Semimartingales
for all f ∈ Pol≤m (S) with m ≥ 2. Furthermore, suppose that (Xt 1{t<T∆ } ) is a
semimartingale, whose characteristics (B, C, ν) with respect to the “truncation
function” χ(ξ) = ξ satisfy the condition of Proposition 4.2.1. If
Px [t < T∆ ] = e−γt
(4.22)
R
for some constant γ ≥ 0 and if Ex [ Rn kξkm Kω,t (dξ)] < ∞, then X is an
m-polynomial process.
Proof. By Condition (4.22) and Theorem 4.1.8, X is 0-polynomial, since
Z t
Z t
γ1{s<T∆ } ds = Px [t < T∆ ] − 1 +
γPx [s < T∆ ] ds
Ex 1{t<T∆ } − 1 +
0
0
Z t
−γt
γe−γs ds = 0.
=e −1+
0
R
Moreover, as a consequence of (4.7) and (4.9), Rn kξkk Kω,t (dξ) < ∞ for all
2 ≤ k ≤ m and t ≥ 0. Hence, for all 2 ≤ l ≤ k ≤ m,
Z t
Z
k−l
l
kXs k 1{s<T∆ }
kξk Kω,s (dξ) ds
(4.23)
Rn
0
t
is of locally integrable variation, as it is a predictable finite variation process.
This implies in particular that (Xt 1{t<T∆ } ) is a special semimartingale and
justifies the choice of the “truncation function” χ(ξ) = ξ. An application of
Itô’s formula to f (x) = xk for 1 ≤ k = |k| ≤ m, and the fact that (4.23) is of
locally integrable variation then implies that
n Z t
X
Di f (Xs )1{s<T∆ } bi (Xs )ds
f (Xt )1{t<T∆ } − f (X0 ) −
i=1
n
X
1
+
2 i,j=1
Z tZ
+
0
0
Z t
Z
Dij f (Xs )1{s<T∆ } cs,ij +
Rn
Rn
0


ξi ξj Kω,s (dξ)
ds

k X
k
Xsk−l 1{s<T∆ } ξ l  Kω,s (dξ)ds
l
|l|=3
(4.24)
is a local martingale. The extended infinitesimal generator G applied to xk is
thus given by the last three terms in equation (4.24), which by our assumptions clearly map Pol≤k (S) into Pol≤k (S) for 1 ≤ k ≤ m. Hence we are in
the situation of Theorem 4.1.8 (iv). The assertion then follows by using the
same arguments as in
R the proof of Theorem 4.1.8 (iv) ⇒ (iii) above and the
assumption that Ex [ Rn kξkm Kω,t (dξ)] < ∞ for all t ≥ 0.
4.3. Characterization by means of the Extended Generator
135
Remark 4.3.3. (i) Let us mention that the characteristics of (Xt 1{t<T∆ } )
in Corollary 4.3.2 and Proposition 4.2.1 are specified with respect to the
“truncation function” χ(ξ) = ξ. While C and ν do not depend on this
choice, the characteristic B does depend on χ. If one chooses another
truncation
function χ0 , then the difference between B and B 0 is given
RtR
by 0 Rn \{0} (χ0 (ξ) − χ(ξ)) ν(ds, dξ). Thus the requirement that C and
ν are as in Corollary 4.3.2 and
X
Z
1
0
(χ(ξ) − χ (ξ)) Kω,t (dξ) =
αk Xtk 1{t<T∆ }
bt 1{t<T∆ } +
Rn \{0}
|k|=0
(4.25)
is an equivalent condition guaranteeing that X is m-polynomial.
(ii) We now give two examples of random measures Kω,t (dξ) which satisfy
the conditions of Proposition 4.2.1 as long as cij ∈ Pol≤2 (S) and b satisfies (4.25).
(a) The first one essentially requires Kω,t (dξ) = K(Xt (ω), dξ) to be a
quadratic polynomial in Xt (ω), that is,
!
n
X
µ00 (dξ) X
µi0 (dξ)
µij (dξ)
+
Xt,i
+
Xt,i Xt,j
,
Kω,t (dξ) =
kξk2 ∧ 1 i=1
kξk2 ∧ 1 i≤j
kξk2 ∧ 1
where all µij are finite signed measures on Rn such that K(x, ·) is
a well-defined Lévy measure for every x ∈ S. Denoting the Jordan
−
decomposition of µij by µ+
ij , µij , it is necessary to require
Z
Z
+
−
m
kξk (µij (dξ) + µij (dξ)) =
kξkm |µij (dξ)| < ∞.
kξk>1
kξk>1
(b) Alternatively, K can be specified as the pushforward of a Lévy measure under an affine function. Let d ≥ 1 and let
g : S × Rd → Rn ,
(x, y) 7→ g(x, y) = g x (y) = H(y)x + h(y),
be an affine function in x. Here, H : Rd → Rn×n and h : Rd → Rn
are assumed to be measurable. We then define K by
Kω,t (dξ) = K(Xt (ω), dξ) := (g Xt )∗ µ(dξ),
where for each x ∈ S, (g x )∗ µ denotes the pushforward of the measure µ under the map g x . Moreover, µ is a Lévy measure on Rd
integrating
Z
kH(y)kk + kh(y)kk µ(dy) for all 1 ≤ k ≤ m.
Rd \{0}
136
4.4
Chapter 4. Characterization and Relation to Semimartingales
Examples
Example 4.4.1 (Affine processes). Every affine process X on S = Rp+ × Rn−p
is m-polynomial if the
R killing rate γ is constant, if the Lévy measures µi ,
i = 1, . . . , p, satisfy kξk>1 kξkµi (dξ) < ∞, and if Pt f (x) is well-defined for
f ∈ Pol≤m (S).
Proof. The characteristic function of an affine process is given by
Ex ehi u,Xt i = eφ(t,i u)+hψ(t,i u),xi ,
where φ and ψ are solutions of ordinary differential equations. The assumptions on the killing rate and the jump measure imply that ψ(t, 0) = 0 for
all t ≥ 0 (see Duffie et al. [2003, Proposition 9.1, Lemma 9.2]), whence X
is 0-polynomial. By our assumptions, moments up to order m exist, which
implies that the characteristic function is m-times continuously differentiable.
A computation shows that for all k ≤ m, the k th derivative is a polynomial in
x of degree k. Continuity of t 7→ Pt f (x) follows from the fact that the derivatives of φ and ψ with respect to u are also solutions of ordinary differential
equations.
Note that the explicit knowledge of φ and ψ is not necessary to compute
the moments of an affine process. Simply the knowledge of its characteristics,
which determine the linear map A, is enough.
Example 4.4.2 (Lévy
processes). Let L be a Lévy process on Rn with triplet
R
(b, c, µ) satisfying kξk>1 kξkm µ(dξ) < ∞. Then the Markov process X = x+L
is m-polynomial.
Example 4.4.3 (Exponential Lévy models). Exponential Lévy models are
of the form X = xeL , where L is a RLévy process on R with triplet (b, c, µ).
Under the integrability assumption |y|>1 emy µ(dy) < ∞, which guarantees
m
the existence of Ex [|X
t | ], exponential Lévy models are m-polynomial, since
we have Ex xm emLt = xm etψ(m) , where ψ denotes the cumulant generating
function of the Lévy process.
In order to apply Theorem 4.1.8 (iv) ⇒ (i) together with Remark 4.3.1 or
Corollary 4.3.2 to the following examples, we assume m ≥ 2, i.e., the process
X admits moments up to order 2 at least.
Example 4.4.4 (Lévy driven SDEs). Let L denote a Lévy process on Rd with
triplet (b, c, µ). Suppose furthermore that V1 , . . . , Vd are affine functions, i.e.,
4.4. Examples
137
we have Vi : S → Rn , x 7→ Hi x + hi , where Hi ∈ Rn×n and hi ∈ Rn . A process
X which solves the stochastic differential equation
dXt =
d
X
Vi (Xt− )dLt,i ,
X0 = x ∈ S,
i=1
and leaves S invariant, is m-polynomial, if t 7→ Pt f (x) is continuous for x ∈ S
and f ∈ Pol≤m (S). This is for instance the case if moment conditions on the
Lévy measure of the type
Z
kykm+ µ(dy) < ∞
(4.26)
kyk>1
hold true for some > 0.
Proof. For C 2 -functions and general Lipschitz continuous functions V1 , . . . , Vd
the extended infinitesimal generator of X with respect to some truncation
function χ is given by
n
1X
((V1 (x) . . . Vd (x))c(V1 (x) . . . Vd (x))0 )ij Dij f (x)
2 i,j=1
* d
+
X
+
Vi (x)bi , ∇f (x)
i=1
Z
+
f
x+
d
X
!
Vi (x)yi
− f (x) −
* d
X
i=1
+!
Vi (x)χi (y), ∇f (x)
µ(dy).
i=1
Concerning the random measure Kω,t (dξ), this example corresponds to the
P
situation of Remark 4.3.3 (ii) (b) with g(x, y) = H(y)x + h(y) = di=1 Hi yi x +
hi yi . The condition of Remark 4.3.1, that is,
Z
Ex
Rn
m
#
d
X
Hi yi Xt + hi yi µ(dy) < ∞,
Rn "Z
kξkm Kω,t (dξ) = Ex
i=1
is satisfied due to (4.26). Moreover, the existence of the (m + )th -moment of
the measure µ, that is, Condition (4.26), is sufficient for Ex [kXt km+ ] to be
finite (see Protter [2005, Theorem V.67] or Jacod et al. [2005]), which then also
implies continuity of t 7→ Pt f (x) for all f ∈ Pol≤m (S). Hence Theorem 4.1.8
(iv) ⇒ (i) and Remark 4.3.1 yield the assertion.
138
Chapter 4. Characterization and Relation to Semimartingales
Example 4.4.5 (Quadratic term structure models Chen, Filipović, and Poor
[2004]). Consider the following quadratic short rate model r, specified as nonnegative quadratic function of a one-dimensional Ornstein-Uhlenbeck process
Y
rt = R0 + R1 Yt + R2 Yt2 ,
for appropriate Ri ∈ R. Here, Y is given by
dYt = (b + βYt )dt + σdWt ,
b, β ∈ R,
σ ∈ R+ ,
where W is a standard Brownian motion. The joint process X = (Y, r) then
follows the dynamics
dYt
b
β
0
=
+
Yt +
rt dt
drt
R1 b + R2 σ 2 − 2R0 β
2R2 b − R1 β
2β
σ
+
dWt .
(R1 + 2R2 Yt )σ
Due to the implication (iv) ⇒ (i) of Theorem 4.1.8, it is thus a polynomial
process with
Z t 1 R1
0
2R2
0 0
2
2
2
Ct =
σ
+σ
Ys + σ
Ys2 ds.
2
2
R
R
2R
4R
R
0
4R
1
2
1 2
1
2
0
Example 4.4.6 (Jacobi process). Another example of a polynomial process is
the Jacobi process (see Gourieroux and Jasiak [2006]), which is the solution
of the stochastic differential equation
p
dXt = −β(Xt − θ)dt + σ Xt (1 − Xt )dWt , X0 = x ∈ [0, 1],
on S = [0, 1], where θ ∈ [0, 1] and β, σ > 0. This example can be extended by
adding jumps, where the jump times correspond to those of a Poisson process
N with intensity λ and the jump size is a function of the process level. Indeed,
if a jump occurs, then the process is reflected at 12 so that it remains in the
interval [0, 1], i.e., the (extended) infinitesimal generator is given by
df (x)
1 2
d2 f (x)
− β(x − θ)
Gf = σ (x(1 − x))
+ λ(f (1 − x) − f (x)).
2
2
dx
dx
By applying Theorem 4.1.8 (iv) ⇒ (i), it is thus easily seen that this process is
polynomial. In terms of Remark 4.3.3 (ii) (b), we have here g(x, y) = −2yx+y
and µ(dy) = λδ1 (dy).
4.4. Examples
139
Example 4.4.7 (Pearson diffusions). Example 4.4.6 (without jumps) as well
as the Ornstein-Uhlenbeck and the Cox-Ingersoll-Ross process, all of them with
mean-reverting drift, can be subsumed under the class of the so-called Pearson
diffusions, which are solutions of SDEs of the form
q
(4.27)
dXt = −β(Xt − θ)dt + (a + α10 Xt + α11 Xt2 )dWt , X0 = x,
where β > 0 and α10 , α11 and a are specified such that the square root is
well-defined. Forman and Sørensen [2008] give a complete classification of
the different types of the Pearson diffusion in terms of their invariant distributions. Note that existence of (weak) solutions of (4.27) follows from continuity and linear growth of the drift and the diffusion part (see Rogers and
Williams [1987, Theorem V.23.5]).
Example 4.4.8 (Dunkl process). The (extended) infinitesimal generator of
the so-called Dunkl process (see Dunkl [1992], Gallardo and Yor [2006]) is
given by
Z λ
df (x)
d2 f (x)
+ 2
δ−2x (dξ)
f (x + ξ) − f (x) − ξ
Gf (x) =
dx2
2x R
dx
d2 f (x) λ df (x) λ(f (−x) − f (x))
=
+
+
.
dx2
x dx
2x2
Since Kω,t (dξ) =
λ
δ
(dξ)
2Xt2 −2Xt
λ| − 2Xt |m
|ξ| Kω,t (dξ) = Ex
<∞
2Xt2
R
Z
Ex
and since
m
for m ≥ 2, we derive from Theorem 4.1.8 and Remark 4.3.1 that the Dunkl
process is a polynomial process.
Part III
Applications
141
Chapter 5
Multivariate Affine Stochastic
Volatility Models
In this chapter we study applications of affine processes on the symmetric cone
of positive semidefinite d × d matrices, denoted by Sd+ and always assume
d > 1. These matrix-valued affine processes have arisen from a large and
growing range of useful applications in finance, including multi-asset option
pricing with stochastic volatility and correlation structures, and fixed-income
models with stochastically correlated risk factors and default intensities. Here,
we focus on multivariate affine stochastic volatility models.
For illustration, let us consider the following model consisting of a ddimensional logarithmic price process with risk-neutral dynamics
p
1 diag
dYt = r1 − Xt
dt + Xt dVt ,
2
Y0 = y,
(5.1)
and stochastic covariation process X = hY, Y i. Here, V denotes a standard ddimensional Brownian motion, r the constant interest rate, 1 the vector whose
entries are all equal to one and X diag the vector containing the diagonal entries
of X.
The necessity to specify X as a process in Sd+ such that it qualifies as
covariation process is one of the mathematically interesting aspects of such
models. Beyond that, the modeling of X must allow for enough flexibility in
order to reflect the stylized facts of financial data and to adequately capture
the dependence structure of the different assets. If these requirements are met,
the model can be used as a basis for financial decision-making in the area of
multi-asset options pricing and hedging of correlation risk.
The tractability of such a model, for example for computing prices of
European derivatives, crucially depends on the dynamics of X. A large part
143
144
Chapter 5. Multivariate Affine Volatility Models
of the literature in the area of multivariate stochastic volatility modeling has
proposed the following affine dynamics for X
p
p
dXt = (b + M Xt + Xt M > )dt + Xt dWt Σ + Σ> dWt> Xt + dJt ,
(5.2)
X0 = x ∈ Sd+ ,
where b is some suitably chosen matrix in Sd+ , M, Σ some invertible matrices,
W a standard d × d matrix of Brownian motions possibly correlated with V ,
and J a pure jump process whose compensator is an affine function of X. This
affine multivariate stochastic volatility model generalizes the well-known onedimensional models of Heston [1993], for the diffusion case, and of BarndorffNielsen and Shephard [2001], for the pure jump case. It is tractable in the
sense that the characteristic function of (X, Y ) is known up to the solution of
an ordinary differential equation.
In the absence of jumps, the process described by (5.2) is a so-called
Wishart process, as its marginal distributions are the non-central Wishart
distributions described in Section 3.5 and Section 3.7. These processes were
first introduced by Bru [1989, 1991] and have subsequently been applied in
mathematical finance.
Gourieroux and Sufana [2003, 2004] and Gourieroux, Monfort, and Sufana
[2005] seem to be the first who considered these processes for term structure
modeling. Other financial applications thereof have then been taken up and
carried further by various authors, including Da Fonseca, Grasselli, and Ielpo
[2007a, 2009], Da Fonseca, Grasselli, and Tebaldi [2007b, 2008], Buraschi, Cieslak, and Trojani [2007] and Buraschi, Porchia, and Trojani [2010]. BarndorffNielsen and Stelzer [2007, 2011] provided a theory for a certain class of matrixvalued Lévy driven Ornstein-Uhlenbeck processes of finite variation. These
processes have then been considered for multivariate stochastic volatility modeling in Pigorsch and Stelzer [2009a] and Muhle-Karbe, Pfaffel, and Stelzer
[2010]. Leippold and Trojani [2008] introduced Sd+ -valued affine jump diffusions and provided financial examples, including multivariate option pricing,
fixed-income models and dynamic portfolio choice.
5.1
Definition
Analogously to the one dimensional case (see Keller-Ressel [2010, Section 2]),
we define a multivariate affine stochastic volatility model via the joint moment generating function of the logarithmic price process and the covariation
process: We assume that the d-dimensional asset price process (St )t≥0 is given
by
St = ert+Yt , t ≥ 0,
5.1. Definition
145
where r is the constant nonnegative interest rate and (Yt )t≥0 the d-dimensional
discounted logarithmic price process starting at Y0 = y ∈ Rd a.s. The discounted price process is thus simply (eYt )t≥0 . Therefore we shall assume in
the sequel that r = 0, and that (St )t≥0 is already discounted. Let (Xt )t≥0
denote the stochastic covariation process, which takes values in Sd+ , the cone
of positive semidefinite d × d matrices and starts at X0 = x ∈ Sd+ a.s. In
order to qualify for a multivariate affine stochastic volatility model, the joint
process (Xt , Yt )t≥0 with state space D := Sd+ × Rd has to satisfy the following
assumptions:
A1) (Xt , Yt )t≥0 is a stochastically continuous time-homogeneous Markov process on D = Sd+ × Rd .
A2) The Fourier-Laplace transform of (Xt , Yt ) has exponential affine dependence on the initial states (x, y), that is, there exist functions (t, u, v) 7→
Φ(t, u, v) and (t, u, v) 7→ Ψ(t, u, v) such that
h
i
>
>
Ex,y etr(uXt )+v Yt = Φ(t, u, v)etr(Ψ(t,u,v)x)+v y
(5.3)
for all (x, y) ∈ D and all (t, u, v) ∈ Q where
n
Q = (t, u, v) ∈ R+ × Sd + i Sd × Cd i
h
i
o
h
tr(uXt )+v> Yt tr(Re(u)Xt )+Re(v)> Yt
<∞ .
Ex,y e
= Ex,y e
Remark 5.1.1. (i) Since we shall establish conditions under which S is a
martingale, we need in particular that Ex,y [eYt,i ] < ∞ for i ∈ {1, . . . , d}.
For this reason we here suppose – in contrast to Definition 1.1.4 – that
the affine property holds on the set Q ⊇ R+ ×U and not only on R+ ×U.
Note that in this case the set U corresponds to Sd− + i Sd × i Rd .
(ii) Due to the term v > y in the moment generating function, we here do not
consider all affine processes on Sd+ × Rd . For stochastic volatility models
it is however reasonable to suppose this form of the moment generating
function. It simply means that Yt with Y0 = y is shifted by c for every
t ≥ 0 if we start at y + c. Note that Assumption A2) also implies that
the covariation process X is a Markov process on Sd+ in its own right
such that we can apply the theory developed in Chapter 3.
(iii) For the moment, we do not make the assumption that (St ) is conservative
or a martingale. Instead this will be established in the following section.
146
Chapter 5. Multivariate Affine Volatility Models
(iv) A possible extension of affine stochastic volatility models, as introduced
above, is to assume that the interest rate r is described by an affine short
rate model (see Duffie et al. [2003, Chapter 11]).
The following theorem summarizes some important properties of multivariate affine stochastic volatility models and is based on the results of Chapter 1.
For the sake of readability, we state the proof nevertheless by using the same
arguments as in Proposition 1.1.6 (iv) and applying Theorem 1.5.4.
Theorem 5.1.2. Let (τ, u, v) ∈ Q for some τ ≥ 0 and suppose that
i
h
tr(uXτ )+v > Yτ
6= 0.
E0,0 e
(5.4)
Then, for t, s ≥ 0 such that t + s = τ , we have (t, u, v) ∈ Q, (s, Ψ(t, u, v), v) ∈
Q and
h
i
h
i
tr(uXt )+v > Yt
tr(Ψ(t,u,v)Xs )+v > Ys
E0,0 e
6= 0, and E0,0 e
6= 0.
(5.5)
Moreover, the functions Φ and Ψ satisfy the semiflow equations, that is,
Φ(t + s, u, v) = Φ(t, u, v)Φ(s, Ψ(t, u, v), v),
Ψ(t + s, u, v) = Ψ(s, Ψ(t, u, v), v),
(5.6)
and the derivatives
∂Φ(t, u, v) F(u, v) :=
∂t
t=0
∂Ψ(t, u, v) and R(u, v) :=
∂t
(5.7)
t=0
exist and are continuous in (u, v). Furthermore, for t ∈ [0, τ ), Φ and Ψ satisfy
the generalized Riccati equations
∂t Φ(t, u, v) = Φ(t, u, v)F(Ψ(t, u, v), v),
∂t Ψ(t, u, v) = R(Ψ(t, u, v), v),
Φ(0, u, v) = 1,
Ψ(0, u, v) = u.
(5.8)
(5.9)
Proof. By the Markov property and the law of iterated expectations, we have
for all t, s ≥ 0 such that t + s = τ
h
i
tr(Ψ(t+s,u,v)x)+v > y
tr(uXt+s )+v > Yt+s
Φ(t + s, u, v)e
= Ex,y e
ii
h
h
>
= Ex,y Ex,y etr(uXt+s )+v Yt+s Fs
h
h
ii
>
= Ex,y EXs ,Ys etr(uXt )+v Yt .
(5.10)
5.2. Form of F and R
147
Hence (t, u, v) ∈ Q and (s, Ψ(t, u, v), v) ∈ Q. Moreover, by Assumption (5.4),
neither the inner nor the outer expectation can be 0, which thus implies (5.5).
We can therefore write (5.10) as
i
i
h
h
>
>
Ex,y etr(uXt+s )+v Yt+s = Ex,y Φ(t, u, v)etr(Ψ(t,u,v)Xs )+v Ys
= Φ(t, u, v)Φ(s, Ψ(t, u, v), v)etr(Ψ(s,Ψ(t,u,v),v)x)+v
>y
for all (x, y) ∈ Sd+ × Rd . Combining this with the left hand side of (5.10),
already implies (5.6).
Differentiability of Φ(t, u, v) and Ψ(t, u, v) at t = 0 and continuity of F(u, v)
and R(u, v) in (u, v) follow from Theorem 1.5.4. Note that the proof works exactly alike for (u, v) ∈
/ U. Due to this property, we are allowed to differentiate
the equations (5.6) with respect to t and evaluate them at 0. As a consequence,
Φ and Ψ satisfy the generalized Riccati equations (5.8) and (5.9).
5.2
Form of F and R
Similar to Section 2.3.1 and 2.3.2, we are interested in the particular form
of the functions F and R and the restrictions on the involved parameters.
From Theorem 1.5.4 it already follows that F and R have parametrization
of Lévy Khintchine type, but in order to obtain the particular admissibility
constraints in this setting, we proceed similarly as in Proposition 2.3.2 and
Proposition 2.3.3. Note that we cannot directly apply the results of Chapter 2
and Chapter 3, since we are here working on the mixed state space Sd+ × Rd .
Before stating the theorem, let us first introduce some notation. We define
the following set
V = {(u, v) ∈ Sd + i Sd × Cd | ∃ t > 0 such that (t, u, v) ∈ Q}.
Note that U ⊆ V. Moreover, for an element in z ∈ Sd × Rd , we write zX for
the components in Sd and zY for those in Rd . In particular, χY : Rd → Rd
denotes some bounded continuous truncation functions.
Theorem 5.2.1. For (u, v) ∈ V, the functions F and R are of Lévy-Khintchine
148
Chapter 5. Multivariate Affine Volatility Models
form, that is,
1
F(u, v) = v > aY v + tr(bX u) + v > bY − c
2Z
>
+
etr(ξX u)+v ξY − 1 − v > χY (ξY ) m(dξ),
(5.11)
D
1
>
(u) + BY> (v) − γ
R(u, v) = 2uαX u + QY (v) + L(u, v) + BX
Z 2
>
etr(ξX u)+v ξY − 1 − v > χY (ξY ) µ(dξ),
+
(5.12)
D
where
αX ∈ Sd+ ,
(5.13)
d
QY : R →
Sd+
is a quadratic function,
d
(5.14)
L : Sd × R → Sd is a bilinear function such that
(5.15)
4uαX u + QY (v) + 2L(u, v) ∈ Sd+ ,
(5.16)
∀(u, v) ∈ Sd × Rd ,
µ is an Sd+ -valued Borel measure with supp(µ) ⊆ D such that, for every
x ∈ Sd+ , M (x, dξ) = tr(µ(dξ)x) is a Lévy measure, which satisfies
Z q
>
2
tr(ξX ) + ξY ξY ∧ 1 M (x, dξ) < ∞,
(5.17)
D
and for all x ∈ Sd+ and (u, v) ∈ V
Z
tr(Re(u)ξX )+Re(v)> ξY
M (x, dξ) < ∞,
n√
oe
D∩
(5.18)
2 )+ξ > ξ >1
tr(ξX
Y Y
>
: Sd → Sd , BY> : Rd → Sd are linear maps such that, for all u ∈ Sd− ,
BX
x ∈ Sd+ with tr(ux) = 0,
>
tr(BX
(u)x) ≤ 0,
(5.19)
and
γ ∈ Sd+ ,
aY ∈ Sd+ ,
(5.20)
(5.21)
bX − (d − 1)αX ∈ Sd+ , bY ∈ Rd ,
c ∈ R+ .
(5.22)
(5.23)
5.2. Form of F and R
149
Finally, m is a Borel measure with supp(m) ⊆ D such that
Z q
>
2
tr(ξX ) + ξY ξY ∧ 1 m(dξ) < ∞,
D
Z
tr(Re(u)ξX )+Re(v)> ξY
m(dξ) < ∞, ∀(u, v) ∈ V.
n√
oe
D∩
(5.24)
(5.25)
2 )+ξ > ξ >1
tr(ξX
Y Y
Proof. We proceed similarly
as in the
i proof of Proposition 2.3.2. Note that
h
tr(uXt )+v > Yt
at t = 0 exists for all (x, y) ∈ D and
the t-derivative of Ex,y e
(u, v) ∈ V, since
i
h
>
tr(uXt )+v > Yt
− etr(ux)+v y
Ex,y e
lim
t↓0
t
>
>
Φ(t, u, v)etr(Ψ(t,u,v),x)+v y − etr(ux)+v y
= lim
t↓0
t
= (F(u, v) + tr(R(u, v)x)etr(ux)+v
>y
is well-defined by Theorem 5.1.2. Moreover, we can also write
(F(u, v) + tr(R(u, v)x)
i
h
>
>
Ex,y etr(uXt )+v Yt − etr(ux)+v y
= lim
t↓0
tetr(ux)+v> y
Z
1
tr(u(ξX −x))+v > (ξY −y)
e
pt (x, y, dξ) − 1
= lim
t↓0 t
D
Z
1
>
= lim
etr(uξX )+v ξY − 1 pt (x, y, dξ + (x, y))
t↓0
t D−(x,y)
!
pt (x, y, D) − 1
+
,
t
where pt (·, ·, dξ) denotes the Markov kernels. By the above equalities and the
fact that pt (x, y, D) ≤ 1, we then obtain for (u, v) = 0
0 ≥ lim
t↓0
pt (x, y, D) − 1
= F(0, 0) + tr(R(0, 0), x).
t
Setting −F(0, 0) = c and −R(0, 0) = γ yields (5.23) and (5.20). We thus have
(F(u, v) + c) + tr((R(u, v) + γ)x)
Z
1
>
etr(uξX )+v ξY − 1 pt (x, y, dξ + (x, y)) (5.26)
= lim
t↓0 t D−(x,y)
150
Chapter 5. Multivariate Affine Volatility Models
By the same arguments as in the proof of Proposition 2.3.2, it follows that
the left hand side of (5.26) is the logarithm of the Fourier-Laplace transform
of some infinitely divisible probability distribution K(x, y, dξ) supported on
(Sd+ − R+ x × Rd ).
In particular, for x = 0, K(0, y, dξ) is an infinitely divisible distribution
with support on D which cannot depend on y. Defining a scalar product on
Sd × Rd by h(u, v), (x, y)i := tr(ux) + v > y, we have by the Lévy–Khintchine
formula
1
F(u, v) + c = h(u, v), a(u, v)i + h(bX , bY ), (u, v)i
2Z
+
eh(ξX ,ξY ),(u,v)i − 1 − h(χX (ξ), χY (ξ)), (u, v)i m(dξ),
Sd ×Rd
where a is a positive semidefinite linear operator on Sd+ ×Rd , bX ∈ Sd , bY ∈ Rd ,
m a Lévy measure with support on Sd × Rd , χX an Sd -valued and χY and Rd valued truncation function. Note that if (u, v) ∈ V, then (Re(u), Re(v)) ∈ V
as well and in particular F(Re(u), Re(v)) is well defined. Therefore
h
i
h(Re(u),Re(v),(LX
,LY
)i
1
1
E e
< ∞.
Here, (LX , LY ) corresponds to the Lévy process associated to K(0, y, dξ). This
in turn implies
Z
eh(ξX ,ξY ),(Re(u),Re(v))i m(dξ) < ∞.
(5.27)
Sd ×Rd ∩{kξk>1}
Let now T denote the projection of Sd+ × Rd on Sd+ . We can then apply
the same arguments as in the proof of Proposition 2.3.3. Indeed, by the LévyKhintchine formula on cones, the logarithm of the Fourier-Laplace transform
of T∗ K(0, y, dξ) is given by
Z
F(u, 0) = tr(bX u) +
etr(ξX u) − 1 m(T −1 (dξX )),
Sd+
where bX ∈ Sd+ and m satisfies supp(T∗ m) ⊆ Sd+ as well as
Z q
2
tr(ξX ) ∧ 1 m(T −1 (dξX )) < ∞,
Sd+
which together with (5.27) yields (5.24) and (5.25). In particular, we can
choose χX to be 0 and χY only to depend on ξY . Moreover, we have T aT > = 0
5.2. Form of F and R
151
and since a is a positive semidefinite operator on Sd+ × Rd , we obtain Condition (5.21).
Using again the same arguments as in the proof of Proposition 2.3.2, we
conclude that exp(tr((R(u, v)+γ)x)) is the Fourier-Laplace transform of some
infinitely divisible distribution L(x, y, dξ) on (Sd+ − R+ x) × Rd . Hence
tr((R(u, v) + γ)x)
1
= h(u, v), A(x)(u, v)i + h(BX (x), BY (x)), (u, v)i
2Z
eh(ξX ,ξY ),(u,v)i − 1 − h(χX (ξ), χY (ξ)), (u, v)i M (x, dξ),
+
Sd ×Rd
where, for each x ∈ Sd+ , A(x) is a positive semidefinite linear operator on
Sd+ × Rd , BX (x) ∈ Sd , BY (x) ∈ Rd , M (x, ·) is a Lévy measure with support
on Sd × Rd , χX an Sd -valued and χY an Rd -valued truncation function. By
the same arguments as before, we have
Z
eh(ξX ,ξY ),(Re(u),Re(v))i M (x, dξ) < ∞.
(5.28)
Sd ×Rd ∩{kξk>1}
We are again interested in the restrictions on the parameters. To this end,
let now U denote the projection of Sd × Rd on Sd . We are then in the situation
of the particular Euclidean Jordan algebra Sd , as discussed in Chapter 3 and
Section 3.7. Using these results, we conclude that the logarithm of the FourierLaplace transform of U∗ L(x, y, ·) is given by
tr((R(u, 0) + γ)x) =
1
tr(u(AX (x)u)) + tr(BX (x)u)
2Z
etr(uξX ) − 1 M (x, U −1 (dξX ))
+
Sd+
(5.29)
>
= 2 tr((uαX u)x) + tr(BX
(u)x)
Z
+
etr(uξX ) − 1 tr(µ(U −1 (dξX ))x).
Sd+
Here, for each x ∈ Sd+ , AX (x) is a positive semidefinite operator on Sd+ ,
depending linearly on x and satisfying tr(u(AX (x)u)) = 4 tr((uαX u)x, where
αX ∈ Sd+ , thus (5.13). Note here that the quadratic representation P (u)αX
on the cone of positive semidefinite matrices is given by P (u)αX = uαX u.
Moreover, due to Proposition 3.2.3 and since the rank d of Sd is supposed to
be greater than 1, we can choose χX (ξ) to be 0. Similarly as before χY (ξ)
can also be chosen to depend only on ξY . By the linearity in x, we can write
152
Chapter 5. Multivariate Affine Volatility Models
M (x, dξ) = tr(µ(dξ)x) for the Lévy measures M (x, ·), where µ is an Sd+ -valued
Borel measure with support on Sd+ × Rd . Concerning integrability, we have
Z q
2
tr(ξX
) ∧ 1 M (x, U −1 (dξX )) < ∞,
Sd+
>
which together with (5.28) implies (5.17) and (5.18). Furthermore, BX
: Sd →
−
+
Sd is a linear map, which satisfies for all u ∈ Sd and x ∈ Sd with tr(ux) = 0
the following inward pointing drift condition
>
tr(BX
(u)x) ≤ 0,
hence implying (5.19). The remaining admissibility conditions follow from the
fact that x 7→ 21 h(u, v), A(x)(u, v)i and x 7→ BY (x)v > are linear maps and A(x)
is positive semidefinite. Hence BY> : Rd → Sd is linear and 21 h(u, v), A(x)(u, v)i
can be written as
1
1
h(u, v), A(x)(u, v)i = tr(Q((u, v)), x),
2
2
where Q : Sd × Rd → Sd+ is a quadratic function. It can be decomposed into
Q((u, v)) = 4uαX u + QY (v) + 2L(u, v),
where QY : Rd → Sd is a quadratic function, that is, Condition (5.14), and
L : Sd × Rd → Sd a bilinear function, that is, Condition (5.15), such that
Q((u, v)) ∈ Sd+ for all (u, v) ∈ Sd × Rd , which implies (5.16). Finally, since for
u ∈ Sd− , F(u, 0) and R(u, 0) correspond to the derivatives of Φ and Ψ of an
affine process on Sd+ , Proposition 3.2.6 implies the drift Condition (5.22).
Remark 5.2.2. Note the following consequences of Condition (5.16). Let
x ∈ Sd+ and u ∈ Sd such that ux = 0. Then
tr(L(u, v)x) = 0,
(5.30)
for all v ∈ Rd . Indeed, in this case tr(4uαX ux) = 0 and thus tr(QY (v)x) +
2 tr(L(u, v)x) ≥ 0 for all v ∈ Rd . Hence
− tr(QY (v)x) ≤ 2 tr(L(u, v)x) ≤ tr(QY (v)x).
Dividing by kvk and letting v → 0, then yields (5.30). Note here that we have
ux = 0 if u ∈ Sd− , x ∈ Sd+ and tr(ux) = 0 is satisfied.
Moreover, if αX or QY is degenerate, we obtain by the same argument that
tr(L(u, v)x) = 0,
(5.31)
5.3. Conservativeness and Martingale Property
153
for all (u, v) ∈ Sd × Rd for which we have tr(4uαX ux) = 0 or tr(QY (v)x) = 0.
In order to give some intuition for Condition (5.30), let us establish a
n−m
. Consider the linear difconnection to the canonical state space Rm
+ ×R
fusion components, denoted by (α1 , . . . , αm ). Then the above condition on
L(u, v) corresponds to the fact that αi,IJ has to satisfy a certain structure
which guarantees that αi is positive semidefinite. We here use the notation
of Duffie et al. [2003, Definition 2.6].
5.3
Conservativeness and Martingale Property
This section is devoted to investigate necessary and sufficient conditions such
that Si = eYi , i = 1, . . . , d, is a martingale. Under such conditions S may
serve as price process under the risk neutral measure in an arbitrage free asset
pricing model. Before addressing this issue, we first clarify when the process
(X, Y ) is conservative. To this end, we apply a result of Mayerhofer et al.
[2011] to our setting. For the notion of quasi-monotonicity, which is needed
here, we refer to Definition 2.3.5
Lemma 5.3.1. The function u 7→ R(u, 0), defined in (5.12), is quasi-monotone increasing on Sd− and locally Lipschitz continuous on Sd−− . Moreover,
for every u ∈ Sd−− , there exists a unique global Sd−− -valued solution Ψ(t, u, 0)
of (5.9).
Proof. The first assertion follows from Proposition 2.3.7 and the second one
from Proposition 2.4.3. Note here that, for u ∈ Sd− , R(u, 0) = −R(−u) and
Ψ(t, u, 0) = −ψ(t, −u).
The following lemma is a reformulation of Mayerhofer et al. [2011, Proposition 3.3].
Lemma 5.3.2. Let T > 0 and u ∈ Sd− . If g(t) : [0, T ) → Sd− is a solution of
dg(t)
= R(g(t), 0),
dt
g(0) = u,
(5.32)
then g(t) Ψ(t, u, 0) for all t < T , where denotes the partial order on Sd
induced by Sd+ .
Proof. The proof relies on Mayerhofer et al. [2011, Corollary A.3]. Choosing
f = R and Df = Sd− , the assumptions of quasi-monotonicity, local Lipschitz continuity and the condition Ψ(t, ·, 0) : Sd−− → Sd−− are satisfied due
to Lemma 5.3.1. By Mayerhofer et al. [2011, Corollary A.3], we thus have
g(t) Ψ(t, u, 0) for all u ∈ Sd−− . Continuity of u 7→ Ψ(t, u, 0) then yields the
assertion.
154
Chapter 5. Multivariate Affine Volatility Models
Proposition 5.3.3. Suppose (X, Y ) satisfies the Conditions A1) and A2).
Then (X, Y ) is conservative if and only if F(0, 0) = 0 and g = 0 is the only
Sd− -valued solution of (5.32) with g(0) = 0. Moreover, each of these statements
implies R(0, 0) = 0.
Proof. Using Lemma 5.3.1 and 5.3.2, we can adapt the proof of Mayerhofer
et al. [2011, Theorem 3.4] to our setting without modification.
Remark 5.3.4. As already mentioned in Section 3.3.3, a sufficient condition
for (X, Y ) to be conservative is c = 0, γ = 0 and
Z
kξkM (x, dξ)(dξ) < ∞, ∀x ∈ Sd+ ,
D∩{kξk≥1}
(see Duffie et al. [2003, Section 9]).
Henceforth, let us assume that (X, Y ) is a conservative process. Then
the martingale property of Si = eYi , i ∈ {1, . . . , d}, can be characterized as
stated in Theorem 5.3.5 below. This is a generalization of Keller-Ressel [2010,
Theorem 2.5.b] and Mayerhofer et al. [2011, Remark 4.5] to the setting of
multivariate affine stochastic volatility models.
Theorem 5.3.5. Suppose (X, Y ) is conservative and satisfies the Conditions
A1) and A2). Let i ∈ {1, . . . , d}. Then Si = eYi is a martingale if and only if
the following conditions hold:
Z
>
eei ξY m(dξ) < ∞,
(5.33)
{|e>
i ξY |>1}
Z
>
eei ξY M (x, dξ) < ∞, ∀x ∈ Sd+ ,
(5.34)
{|e>
i ξY |>1}
Z 1 >
>
>
eei ξY − 1 − e>
χ
(ξ
)
m(dξ) = 0, (5.35)
F(0, ei ) = ei aY ei + ei bY +
Y
i Y
2
Z 1
>
e>
ξY
>
i
R(0, ei ) = QY (ei ) + BY (ei ) +
e
− 1 − ei χY (ξY ) µ(dξ) = 0,
2
(5.36)
and g = 0 is the only Sd− -valued solution of
dg(t)
= R∗ (g(t), 0),
dt
where, for u ∈ Sd− , R∗ (u, 0) is defined by
∗
R (u, 0) = 2uαX u + L(u, ei ) +
>
BX
(u)
g(0) = 0,
Z
+
(5.37)
>
etr(ξX u) − 1 eei ξY µ(dξ).
Here, ei , i = 1, . . . , d, denotes ith canonical basis vector of Rd .
5.3. Conservativeness and Martingale Property
155
>
Proof. Let us first suppose that Si = eYi is a martingale. Then Ex,y [eei Yt ] < ∞
and (t, 0, ei ) ∈ Q for all t ∈ R+ . Thus, by Theorem 5.2.1, Conditions (5.25)
and (5.18) are satisfied, which yield (5.33) and (5.34). Moreover, since
h > i
>
>
eei y = Ex,y eei Yt = Φ(t, 0, ei )etr(Ψ(t,0,ei )x)+ei y ,
we have Φ(t, 0, ei ) = 1 and Ψ(t, 0, ei ) = 0. Equations (5.8) and (5.9) thus
imply F(0, ei ) = 0 and R(0, ei ) = 0. The latter then yields for u ∈ Sd−
1
>
(u) + BY> (ei )
R(u, ei ) = 2uαX u + QY (ei ) + L(u, ei ) + BX
2
Z >
etr(ξX u)+ei ξY − 1 − e>
χ
(ξ
)
µ(dξ)
+
Y
i Y
Z
>
>
= 2uαX u + L(u, ei ) + BX (u) +
etr(ξX u) − 1 eei ξY µ(dξ)
Z 1
>
e>
ξY
>
i
e
− 1 − ei χY (ξY ) µ(dξ)
+ QY (ei ) + BY (ei ) +
2
= R∗ (u, 0) + R(0, ei ) = R∗ (u, 0).
Moreover, R∗ (u, 0) is of form (5.12) with corresponding parameters
∗
= αX ,
αX
>∗
>
BX
(u) = BX
(u) + L(u, ei ),
>ξ
Y
µ∗ (dξ) = eei
µ(dξ).
In particular, these parameters satisfy the conditions of Theorem 5.2.1, that
is, (5.13), (5.17) and (5.19). Concerning the latter, we have by (5.19) and (5.30)
>∗
>
(u)x) = tr(BX
(u)x) + tr(L(u, ei )x) ≤ 0,
tr(BX
for all u ∈ Sd− with tr(ux) = 0. Thus the assertions of Lemma 5.3.1 hold true
for R∗ (u, 0). Let now g be a (local) solution of (5.37) on some interval [0, T ),
satisfying g(0) = 0 and taking values in Sd− . By Lemma 5.3.2, we therefore
have Ψ∗ (t, 0, 0) g(t) for all t < T , where
∂Ψ∗ (t, u, 0)
= R∗ (Ψ∗ (t, u, 0), 0),
∂t
Ψ∗ (0, u, 0) = u.
As R∗ (u, 0) = R(u, ei ), we conclude that Ψ∗ (t, 0, 0) = Ψ(t, 0, ei ) = 0, which
yields g = 0.
Concerning the other direction, we can argue as in Kallsen and MuhleKarbe [2010, Corollary 3.4]. Indeed, eYi is a σ-martingale due to (5.33), (5.34),
156
Chapter 5. Multivariate Affine Volatility Models
(5.35) and (5.36). This is a consequence of Kallsen [2003, Lemma 3.1] and the
fact that eYi is a semimartingale, whose characteristics can be determined as
stated in Kallsen [2006, Proposition 3]. Moreover, from Kallsen [2003, Proposition 3.1] it follows that it is a supermartingale, thus it is in particular integrable. Hence (t, 0, ei ) ∈ Q for all t ∈ R+ , and Φ(t, 0, ei ) = 1 and Ψ(t, 0, ei ) = 0
would guarantee that eYi is a martingale. Let now g(t) := Ψ∗ (t, 0, 0) = 0
be the only solution of (5.37). Since (5.36) implies R(u, ei ) = R∗ (u, 0) as
shown above, we have Ψ∗ (t, 0, 0) = Ψ(t, 0, ei ) = 0 and Φ(t, 0, ei ) = 1 follows
from (5.35).
We are now prepared to give our full definition of a multivariate affine
stochastic volatility model. For this purpose we add two further assumptions
to A1) and A2).
A3) The process (X, Y ) is conservative.
A4) The discounted price processes Si = eYi , i = 1, . . . , d, are martingales.
Definition 5.3.6. The process (X, Y ) is called a multivariate affine stochastic
volatility model if it satisfies the Assumptions A1 – A4.
5.4
Semimartingale Representation of Affine
Volatility Models
In order to address the issue of existence and uniqueness of affine stochastic
volatility models, the aim of this section is to establish a semimartingale representation of multivariate affine stochastic volatility models. In particular,
we want to construct a Brownian motion such that (X, Y ) becomes a solution
of a generalized SDE with jumps. To this end, we first study the functions
QY and L, appearing in (5.16).
Lemma 5.4.1. (i) Let QY : Rd → Sd+ be a quadratic function. Then QY (v)
is given by
X
QY (v) =
Gkl vk vl ,
(5.38)
k,l
where Gkl = Glk ∈ Sd , k, l ∈ {1, . . . , d} such that QY (v) ∈ Sd+ for all
v ∈ Rd . In other words, there exists an endomorphism G : Sd+ → Sd+
such that
QY (v) = G(v ⊗ v) = G(vv > ).
5.4. Semimartingale Representation of Affine Volatility Models
157
(ii) Let L : Sd × Rd → Sd be a bilinear function such that (5.30) is satisfied.
Then L is necessarily of the form
!
!
d
d
X
X
L(u, v) = u
Hi vi +
Hi> vi u,
(5.39)
i=1
i=1
where Hi , i = 1, . . . d, are matrices in Rd×d .
Proof. It is clear that any quadratic function with values in Sd+ is of form (5.38).
eY : Rd × Rd → Sd via polarization,
Define now a symmetric bilinear function Q
that is,
eY (v, w) = 1 (QY (v + w) − QY (v) − QY (w)).
Q
2
eY induces a linear map G : Rd ⊗ Rd → Sd such that Q
eY (v, w) =
Then Q
d
d
G(v ⊗ w). Identifying R ⊗ R with the vector space of d × d matrices, yields
the assertion.
e : Sd → Sd such that
Concerning the second statement, consider a map L
e
tr(L(u)x)
= 0 for all u, x ∈ Sd+ with tr(ux) = 0.
(5.40)
e
e
Denote by eLt (u) the semigroup induced by ∂t g(t, u) = L(g(t,
u)). Then (5.40)
e
e
+
Lt
−Lt
implies that e and e
are semigroups which map Sd into Sd+ and thus
e
eLt (Sd+ ) = Sd+ for all t ≥ 0. By Pigorsch and Stelzer [2009b, Theorem 3.3],
e is necessarily of the form L(u)
e
L
= uH + H > u for some matrix H ∈ Rd×d .
e
e with the above property,
Since, for fixed v, we have L(u)
= L(u, v) for some L
the linearity of v 7→ L(u, v) then yields (5.39).
Lemma 5.4.2. Let L be of form (5.39) for some matrices Hi ∈ Rd×d , i =
1, . . . d. Then
u2 + vv > + L(u, v)
(5.41)
is positive semidefinite for all (u, v) ∈ Sd × Rd if and only if all entries of Hi
except the ith column are zero, that is,


0 · · · ρ1 · · · 0


Hi =  ... . . . ... . . . ...  ,
(5.42)
0 · · · ρd · · · 0
where ρ = (ρ1 , . . . , ρd )> with |ρi | ≤ 1 and ρ> ρ ≤ 1. Thus (5.41) can be written
as
u2 + vv > + uρv > + vρ> u.
(5.43)
158
Chapter 5. Multivariate Affine Volatility Models
Proof. Let us first show that positivity of (5.41) for all (u, v) ∈ Sd ×Rd implies
the particular form of the matrices Hi , i ∈ {1, . . . , d}. We first set v = ei ,
where ei denotes the ith basis vector. By choosing u = ±ei e>
i , it is easily
ij
ii
seen that Hi = 0 for all j 6= i, and |Hi | ≤ 1. Taking u = ±εek e>
k for
k 6= i and letting ε → 0, then yields Hikj = 0 for all j 6= i and |Hiki | ≤ 1
for all k ∈ {1, . . . , d}. This already implies the above form of Hi for some
vector ρi = (Hi1i , . . . , Hidi )> . In order to prove that ρ1 = ρ2 = · · · = ρd ,
choose v = aei + bej and u = εId for some a, b ∈ R and ε > 0. By positive
semidefiniteness, we have
((ε2 Id + vv > + ε((Hi + Hi> )a + (Hj + Hj> )b)ij )2
= (ab + εaHiji + εbHjij )2
≤ (ε2 Id + vv > + ε((Hi + Hi> )a + (Hj + Hj> )b)ii
× (ε2 Id + vv > + ε((Hi + Hi> )a + (Hj + Hj> )b)jj
= (ε2 + a2 + 2εaHiii )(ε2 + b2 + 2εbHjjj ).
Dividing by ε and letting ε → 0, this yields
a2 b(Hjjj − Hiji ) + ab2 (Hiii − Hjij ) ≥ 0.
With an appropriate choice of a and b we therefore obtain Hiii = Hjji and
Hjjj = Hiij . Since i, j was arbitrary, the above form of Hi , i = 1, . . . d, for
some vector ρ is proved and we can write
u2 + vv > + L(u, v) = u2 + vv > + uρv > + vρ> u.
It remains to show that ρ> ρ ≤ 1. Choosing v = −ρ and u = Id leads to
Id − ρρ> 0.
(5.44)
Since the only non-zero eigenvalue of ρρ> is given by ρ> ρ, (5.44) is equivalent
to ρ> ρ ≤ 1.
The other direction is obvious, since
u2 + vv > + uρv > + vρ> u = (uρ + v)(uρ + v)> + u(Id − ρρ> )u 0,
by (5.44).
Remark 5.4.3. Note that the assertion of Lemma 5.4.2 also holds true when
replacing u ∈ Sd by u ∈ Rd×d . In this case, we extend the bilinear function L
to Rd×d × Rd in the obvious way by setting
!
!
d
d
X
X
e v) = u
L(u,
Hi vi +
Hi> vi u>
(5.45)
i=1
e v) instead.
and consider uu> + vv > + L(u,
i=1
5.4. Semimartingale Representation of Affine Volatility Models
159
Corollary 5.4.4. Consider the quadratic function given by (5.16), that is,
Q((u, v)) = 4uαX u + QY (v) + 2L(u, v).
Assume that αX is given by αX = Σ> Σ for some Σ ∈ Rd×d and let QY :
Rd → Sd+ be a quadratic function such that QY (v) = G(vv > ), where G is an
endomorphism of Sd+ of the form G : Sd+ → Sd+ , a 7→ G(a) = gag > for some
d × d matrix g. Then
Q((u, v)) = 4uΣ> Σu + gvv > g > + 2L(u, v)
is positive semidefinite for all (u, v) ∈ Sd × Rd if and only if L(u, v) is of the
form
L(u, v) = uΣ> ρv > g > + gvρ> Σu,
(5.46)
for some ρ ∈ Rd with |ρi | ≤ 1 and ρ> ρ ≤ 1.
Proof. By Remark 5.2.2 and Lemma 5.4.1 (ii), positivity of Q((u, v)) implies
that L is of form (5.39) for some matrices Hi , i = 1, . . . , d. We now set
u
e = 2uΣ> and ve = gv and suppose for the moment that Σ and g are invertible.
Then
!
!
d
d
X
X
2L(u, v) = 2u
Hi vi + 2
Hi> vi u
=u
e
i=1
d
X
e i vei
H
!
+
i=1
ei =
where H
Pd
k=1
i=1
d
X
!
e i> vei
H
e u, ve),
u
e> =: L(e
i=1
−1 −>
gki
Σ Hk . Therefore
e u, ve).
Q((u, v)) = u
eu
e> + veve> + L(e
By Remark 5.4.3 and Lemma 5.4.2, this expression is positive semidefinite for
e i , i = 1, . . . , d is of form (5.42) for some ρ ∈ Rd with
all (e
u, ve) ∈ Rd×d × Rd if H
e u, ve) = u
|ρi | ≤ 1 and ρ> ρ ≤ 1. We therefore have L(e
eρe
v > +e
v ρ> u
e> . Substituting
back thus yields (5.46). If Σ and g are degenerate, the necessary direction is
a consequence of (5.31). The sufficient direction follows for general Σ and g
as in the proof of Lemma 5.4.2, since
4uΣ> Σu + gvv > g > + 2uΣ> ρv > g > + 2gvρ> Σu =
(2uΣ> ρ + gv)(2uΣ> ρ + gv)> + 4uΣ> (Id − ρρ> )Σu 0.
160
Chapter 5. Multivariate Affine Volatility Models
We are now prepared to prove the announced semimartingale representation of multivariate stochastic volatility models.
Theorem 5.4.5. Let (X, Y ) be a multivariate affine stochastic volatility model
with F and R of the form (5.8) and (5.9) such that αX = Σ> Σ for some d × d
matrix Σ. Moreover, suppose that QY : Rd → Sd+ is given by QY (v) = gvv > g >
for some d × d matrix g. Then there exist a vector ρ ∈ Rd satisfying |ρi | ≤ 1
and ρ> ρ ≤ 1 and – possibly on an enlargement of the probability space – a d×d
matrix of Brownian motions W , and Rd -valued Brownian motions V and Z,
all mutually independent, such that (X, Y ) admits the following representation
Rt √
√ !
X Xs dWs Σ + Σ> dWs> X
Xt
x
Bt
0
s
Rt √
√
+
=
+
Y
>
e
Bt
Yt
y
aY dZs + g Xs dVs
0
Z tZ 0
+
(µ(X,Y ) (ds, dξ) − ν(ds, dξ))
χ
(ξ
)
Y
Y
0
D
Z tZ ξX
+
µ(X,Y ) (ds, dξ),
ξ
−
χ
(ξ
)
Y
Y
Y
0
D
where
BtX
Z
t
=
(bX + BX (Xs )) ds,
Z t
Z
1
Y
ξY,i
Bt,i = −
aY,ii +
e − 1 − χY,i (ξY ) m(dξ) ds
2
0
D
Z t
1 >
(g Xs g)ii ds
−
0 2
Z tZ
−
eξY,i − 1 − χY,i (ξY ) M (Xs , dξ)ds,
p0 D
e
Vt = ( 1 − ρ> ρ)Vt + Wt ρ,
ν(dt, dξ) = (m(dξ) + M (Xt , dξ))dt,
0
and µX,Y denotes the random measure associated with the jumps of (X, Y ).
Proof. Due to Definition 5.3.6, the functions F and R have to fulfill the conditions of Theorem 5.3.5. Furthermore, as αX = Σ> Σ and QY is given by
QY (v) = gvv > g > , it follows from Corollary 5.4.4 that there exists some ρ ∈ Rd
satisfying |ρi | ≤ 1 and ρ> ρ ≤ 1 such that the positive semidefinite quadratic
function Q((u, v)) defined in (5.16) is of form
Q((u, v)) = 4uΣ> Σu + gvv > g > + 2uΣ> ρv > g > + 2gvρ> Σu.
5.4. Semimartingale Representation of Affine Volatility Models
161
As (X, Y ) is conservative, Theorem 1.4.8 and Theorem 1.5.4 thus imply that
(X, Y ) is a semimartingale, whose characteristics (B, C, ν) with respect to the
truncation function (0, χY ) are given by
Z
t
hBt , (u, v)i =
tr (u (bX + BX (Xs ))) ds
Z t 1
1
>
>
diag(aY ) + diag(g Xs g) ds
−
v
2
2
0
Z tZ
v > eξY − 1 − χY (ξY ) m(dξ)ds
−
Z0 t Z
v > eξY − 1 − χY (ξY ) M (Xs , dξ)ds,
−
Z t 0
v > aY v + 4 tr(uΣ> ΣuXs ) + tr(gvv > g > Xs )
h(u, v), Ct (u, v)i =
0
+ 2 tr(uΣ> ρv > g > Xs ) + 2 tr(gvρ> ΣuXs ) ds,
(5.47)
0
ν(dt, dξ) = (m(dξ) + M (Xt , dξ))dt.
Here, h(u, v), (x, y)i = tr(ux) + v > y, diag(a) is the vector consisting of the
diagonal elements of a matrix a, 1 is the vector whose entries are all equal to
1 and eξY = (eξY,1 , . . . , eξY,d )> . The canonical semimartingale representation
(see Jacod and Shiryaev [2003, Theorem II.2.34]) is thus given by
Xt
Yt
X c Xt
x
Bt
+
=
+
Y
Ytc
Bt
y
Z tZ 0
+
(µ(X,Y ) (ds, dξ) − ν(ds, dξ))
χ
(ξ
)
Y
X
0
D
Z tZ ξX
+
µ(X,Y ) (ds, dξ),
ξ
−
χ
(ξ
)
Y
Y
Y
0
D
where (X c , Y c ) denotes the continuous martingale part. From Theorem 3.7.2
it follows that there exists a d × d matrix of Brownian motions W such that
X c can be written as
Z tp
p
c
Xt =
Xs dWs Σ + Σ> dWs> Xs .
0
In particular,
c
it = Xt,ik (Σ> Σ)jl + Xt,il (Σ> Σ)jk + Xt,jk (Σ> Σ)il + Xt,jl (Σ> Σ)ik .
dhXijc , Xkl
162
Chapter 5. Multivariate Affine Volatility Models
Thus it remains to establish the representation
Ytc
Z t
p
√
>
e
=
aY dZs + g
Xs dVs =: Lt .
0
To this end, note that the quadratic variation of the right hand side satisfies
dhLi , Lj it = (aY + g > Xt g)ij
and
dhXijc , Lk it = (g > Xt )ki (Σ> ρ)j + (g > Xt )kj (Σ> ρ)i .
On the other hand, we can write Ct defined in (5.47) as an (d2 + d) × (d2 + d)
matrix
CtX
CtXY
,
(CtXY )> CtY
where the entries are given by
X
c
Ct,(ij)(kl)
= dhXijc , Xkl
it
= Xt,ik (Σ> Σ)jl + Xt,il (Σ> Σ)jk + Xt,jk (Σ> Σ)il + Xt,jl (Σ> Σ)ik ,
XY
Ct,(ij)k
= dhXijc , Lk it
= (g > Xt )ki (Σ> ρ)j + (g > Xt )kj (Σ> ρ)i ,
Y
Ct,ij
= dhLi , Lj it
= (aY + g > Xt g)ij ,
which proves the assertion.
Remark 5.4.6. (i) Note that the only restriction that we impose in Theorem 5.4.5 on the parameters, appearing in the equations (5.8) and (5.9)
for F and R, is to require that QY (v) = G(vv > ), where G is a congruence
map, i.e., an endomorphism of Sd+ of the particular form G(a) = gag >
for some d × d matrix g. Indeed, for d > 2, it is an unsolved problem
to characterize all linear operators mapping Sd+ into Sd+ but not onto
(see, e.g., Li and Pierce [2001], Schneider [1965]). In the case d = 3,
a very simple example of such an endomorphism which cannot be represented as a sum of congruence maps is given in Choi [1975, Theorem
2]. Note however that all automorphisms of Sd+ are congruence maps
with g an invertible d × d matrix. Since aY + gXg > corresponds to the
quadratic covariation of Y , a restriction to automorphisms means that
the volatility depends on all components of X.
5.5. Examples
163
(ii) From the above semimartingale representation it is easily seen that the
covariation process does not depend on Y . In particular, it is an affine
process (in its own filtration) on Sd+ . Theorem 3.7.1 and Theorem 3.7.2
thus imply existence and uniqueness of X. This then also yields existence
and uniqueness of the log-price process, since the dynamics of Y only
depend on X.
5.5
Examples
In this section we study two particular multivariate affine stochastic volatility
models which have been considered in the literature. These examples are special cases of the model described in Theorem 5.4.5. This result now allows us
to read off the necessary conditions for a particular model, like for example the
correlation structure between the Brownian motions or conditions on the drift
parameter. We start with an analog of the Heston model in the multivariate
setting.
5.5.1
Multivariate Heston Model
The dynamics of the multivariate Heston models read as follows,
p
p
dXt = (bX + BX (Xt ))dt + Xt dWt Σ + Σ> dWt> Xt ,
p
1
dYt = − diag(Xt )dt + Xt dVet ,
2
(5.48)
where the parameters satisfy the conditions of Theorem 5.2.1 and 5.3.5. In
the terminology of Theorem 5.2.1, αX = Σ> Σ and QY (v) = vv > . Moreover,
W and Ve are Brownian motions as specified in Theorem 5.4.5. A version of
this model was introduced in Da Fonseca et al. [2007b] under more restrictive
parameter assumptions. Indeed, therein bX = kΣ> Σ with k > d − 1 and
BX (Xt ) = M Xt + Xt M > for some d × d matrix M . The correlation structure
between Ve and W is the same. The restriction bX = kΣ> Σ often leads to
problems when calibrating to data, since the values obtained for k by an
optimization procedure are often smaller than d − 1, which however means
that the process X does not exist. This isssue can be overcome by allowing
for a more general constant drift.
When assuming the particular linear drift BX (Xt ) = M Xt + Xt M > , the
164
Chapter 5. Multivariate Affine Volatility Models
solutions of the associated Riccati equations
∂Φ(t, u, v)
= Φ(t, u, v) tr(bX Ψ(t, u, v)),
∂t
∂Ψ(t, u, v)
1
= 2Ψ(t, u, v)Σ> ΣΨ(t, u, v) + v > v
∂t
2
>
>
+ Ψ(t, u, v)(Σ ρv + M ) + (vρ> Σ + M > )Ψ(t, u, v)
d
−
1X
vi (ei e>
i ),
2 i=1
are given by
Ψ(t, u, v) = (uΨ12 (t, v) + Ψ22 (t, v))−1 (uΨ11 (t, v) + Ψ21 (t, v)),
Z t
tr(bX Ψ(s, u, v))ds ,
Φ(t, u, v) = exp
0
where
Ψ11 (t, v) Ψ12 (t, v)
Ψ21 (t, v) Ψ22 (t, v)
= exp t
> >+M
−2Σ> Σ
Σ ρv
P
d
1
>
>
−(vρ> Σ + M > )
i=1 vi (ei ei )
2 vv −
!
,
(see Da Fonseca et al. [2007b, Proposition 3.3] for details).
As shown in Theorem 5.4.5, the Heston model can be extended by incorporating jumps, both in the covariation and the log-price process. This then
yields generalizations of the one-dimensional so-called Bates model.
5.5.2
Multivariate Barndorff-Nielsen-Shepard Model
A multivariate extension of the Barndorff-Nielsen-Shepard model was introduced in Pigorsch and Stelzer [2009a] under the name multivariate stochastic
volatility model of OU-type. This model has been analyzed in BarndorffNielsen and Stelzer [2011], while calibration related results can be found
in Muhle-Karbe et al. [2010].
Here, the covariation process and the log-price process are described by
dXt = (M Xt + Xt M > )dt + dLt ,
(5.49)
Z
p
1
dYt = − diag(Xt )dt − (eξY − 1)m(dξ) + Xt dVt + P (dLt ),
2
D
where M ∈ Rd×d and P : Rd×d → Rd some linear map. Moreover, V is an Rd valued Brownian motion and L denotes a drift-less matrix subordinator (see,
5.5. Examples
165
e.g., Barndorff-Nielsen and Stelzer [2007]), which is independent of V . Its
triplet is given by (0, 0, U∗ m(dξ)), where U denotes the projection of Sd+ × Rd
on Sd+ and m corresponds to the Lévy measure of (L, P (L)). We write eξY for
(eξY ,1 , . . . , eξY ,d )> and 1 stands for the vector whose entries are all equal to 1.
Here, the truncation function χY can assumed to be 0.
The covariation process X thus follows a positive semidefinite process
of OU-type, which was introduced in Barndorff-Nielsen and Stelzer [2007].
Note that, as before, the linear drift is specified to be of the particular form
BX (Xt ) = M Xt + Xt M > , which allows to keep analytic tractability when
solving the associated Riccati equations, given by
Z
∂Φ(t, u, v)
= Φ(t, u, v) −
v > (eξY − 1)m(dξ)
∂t
D
Z >
etr(ξX Ψ(t,u,v))+v ξY − 1 m(dξ) ,
+
D
d
∂Ψ(t, u, v)
1
1X
= v > v + Ψ(t, u, v)M + M > Ψ(t, u, v) −
vi (ei e>
i ).
∂t
2
2 i=1
Observe that the equation for Ψ is linear and can thus be solved explicitly,
while the solution of Φ can be obtained via integration (see, e.g., Muhle-Karbe
et al. [2010, Theorem 2.5]).
The relation to the one dimensional Barndorff-Nielsen-Shepard model becomes clear when analyzing the marginal dynamics of the one dimensional
log-prices. By Muhle-Karbe et al. [2010, Theorem 2.3] (see also BarndorffNielsen and Stelzer [2011, Proposition 4.3]), we have
f idi
Z
t
(Yt,i ) =
0
1
− Xs,ii ds − t
2
Z
(eξY,i − 1)m(dξ)
D
!
Z t
p
+
Xs,ii dZs,i + Pi (dLs )
,
0
f idi
where = denotes equality for all finite dimensional distributions. If M is
assumed to be diagonal and P : x 7→ (p1 x11 , . . . , pd xdd ) for some vector p ∈ Rd ,
then the model for the ith asset is equivalent in distribution to a univariate
Barndorff-Nielsen-Shepard model, since every diagonal element of the matrix
subordinator L is obviously a univariate subordinator.
Chapter 6
Applications of Polynomial
Processes
The applications of polynomial processes range from statistical GMM estimation procedures to techniques for option pricing and hedging. For instance, the
efficient and easy computation of moments can be used for variance reduction
techniques in Monte-Carlo methods.
As we have seen in Section 4, polynomial processes have the property that
the expressions for all finite moments are analytically known (up to a matrix
exponential). In other words, there is a large subset of claims where the
prices and hedging ratios (Greeks) are explicitly known. Large can be made
precise in the following sense: if the law of XT , say µ, is characterized by its
moments, then “large” means dense, i.e., polynomial claims are dense (with
respect to the L1 (µ) norm) in the set of “all” claims. If this is not the case,
the payoff function can at least be uniformly approximated by polynomials on
some interval, which can be chosen according to the support of the probability
distribution. The explicit knowledge of prices of polynomial claims then allows
to apply variance reduction techniques for Monte-Carlo computations.
6.1
Moment Calculation
By Theorem 4.1.8 we know that there exists a linear map A such that moments
of m-polynomial processes can simply be calculated by computing etA . Indeed,
by choosing a basis he1 , . . . , eN i of Pol≤m (S) the matrix corresponding to this
linear map, which we also denote by A = (Akl )k,l=1,...,N , can be obtained
P
PN
through Aek = N
l=1 Akl el . Writing f as
k=1 αk ek , we then have
Pt f = (α1 , . . . , αN )etA (e1 , . . . , eN )0 ,
167
(6.1)
168
Chapter 6. Applications of Polynomial Processes
which means that moments of polynomial processes can be evaluated simply
by computing matrix exponentials.
By means of the one-dimensional Cox-Ingersoll-Ross process
p
dXt = (b + βXt )dt + σ Xt dWt , b, σ ∈ R+ , β ∈ R,
we exemplify how moments of order m can be calculated. Its (extended)
infinitesimal generator generator is given by
d2 f (x)
1
df (x)
.
Af (x) = σ 2 x
+
(b
+
βx)
2
dx2
dx
Applying A to (x0 , x1 , . . . , xm ) yields the following (m + 1) × (m + 1) matrix


0
...
 b

β
0
...


 0 2b + σ 2

2β
0
.
.
.


2
A= 0
.
0
3b
+
3σ
3β
0
.
.
.




..
.


m(m−1) 2
0
...
mb + 2 σ mβ
Hence Ex (Xt )k = Pt xk = (0, . . . , 1, . . . , 0)etA (x0 , . . . , xk , . . . , xm )0 .
Remark 6.1.1. (i) Note that A is a lower triangular matrix, whose eigenvalues are the diagonal elements. Since in this case they are all distinct,
the matrix is diagonalizable. Of course, there are many efficient algorithms to evaluate such matrix exponentials (see, e.g., Golub and Van
Loan [1996], Moler and Van Loan [1978]).
(ii) For n > 1, for example in the case of the Heston model, similar closed
form expressions can be obtained.
6.1.1
Generalized Method of Moments
In view of this fast technique of moment calculation for polynomial processes,
the Generalized Method of Moments (GMM) qualifies for parameter estimation and thus also for model calibration. The implementation of a typical
moment condition of the type
 n m

m1
Xt 1 Xt+s1 − Ex [Xtn1 Xt+s
]


..
f (Xt , θ) = 
 , ni , mi ∈ N, 1 ≤ i ≤ q,
.
nq mq
nq mq
Xt Xt+s − Ex [Xt Xt+s ]
6.2. Pricing - Variance Reduction
169
m
where θ is the set of parameters to be estimated, is simple since Ex Xtn Xt+s
=
m
n
Ex [Xt EXt [Xs ]] can also be computed easily. In the case of one-dimensional
jump-diffusions, Zhou [2003] has already used this method for GMM estimation.
6.2
Pricing - Variance Reduction
The fact that moments of polynomial processes are analytically known also
gives rise to efficient techniques for pricing and hedging.
Let X be an m-polynomial process and G : S → Rn a deterministic bimeasurable map such that the discounted price processes are given through
St = G(Xt ) under a martingale measure. Typically G = exp if X are logprices. We denote by F = φ(ST ) a bounded measurable European claim for
some maturity T > 0, whose (discounted) price at t ≥ 0 is given by the risk
neutral valuation formula
pFt = Ex φ(ST )Ft = EXt [(φ ◦ G)(XT )] .
Obviously, claims of the form
F = f ◦ G−1 (ST )
(6.2)
for f ∈ Pol≤m (S) are analytically tractable, since we have
pFt = Ex (f ◦ G−1 )(ST )Ft = PT −t f (G−1 (St )) = e(T −t)A f (G−1 (St ))
for 0 ≤ t ≤ T , where A is the previously defined linear map on Pol≤m (S).
Although claims are in practice not of form (6.2), the explicit knowledge of the
price of polynomial claims can be used for variance reductionP
techniques based
1
F
on control variates. Instead of using the estimator π0 = L Li=1 (φ ◦ G)(XTi )
in a Monte-Carlo simulation, where XT1 , . . . , XTL are L samples of XT , we can
use
L
1X
F
(φ ◦ G)(XTi ) − f (XTi ) − Ex [f (XT )] ,
π̂0 =
L i=1
where f ∈ Pol≤m (S) is an approximation of φ◦G and serves as control variate.
Both estimators
are unbiased and the second clearly outperforms the first since
F
Var π̂0 < Var π0F , where the ratio of the variances depends on the accuracy
of the polynomial approximation.
It is worth mentioning that the previous pricing algorithm has also important consequences for hedging, since the Greeks for “polynomial claims”
F = f (XT ) can also be calculated explicitly and efficiently: The coefficients
170
Chapter 6. Applications of Polynomial Processes
of the polynomial x 7→ Ex [f (XT )] can be computed using matrix exponentiation, taking derivatives of this polynomial is then a simple algebraic operation.
To be more precise, the sensitivities of the price process with respect to the
factors X can be calculated by
∇pFt = ∇PT −t f (G−1 (St ))∇G−1 (St ).
Assuming a complete market situation, the claim φ(ST ) = φ ◦ G(XT ) can be
replicated by a trading strategy η, i.e.,
Z T
ηt dSt .
φ(ST ) = Ex [φ(ST )] +
0
Similarly, the polynomial claim f (Xt ) can be replicated by the delta-hedging
strategy ∇pFt and we conclude that
Z
φ(ST ) − f (XT ) = Ex [φ(ST )] − Ex [f (XT )] +
T
(η − ∇pFt )dSt .
0
If we assume that φ(ST ) − f (XT ) has small variance, then also the stochastic
integral representing the difference of the cumulative gains and losses of the
two hedging portfolios, namely the one built by the unknown strategy η and
the one built by the known strategy ∇pFt , is small.
Concerning the approximation of φ ◦ G by a polynomial, let us consider
the case St = G(Xt,1 ) with G : R → R+ , meaning that we only have one
asset which depends on the first component of the polynomial process X as it
is usually the case in stochastic volatility models. If the Hamburger moment
problem (see, e.g., Reed and Simon [1975]) for the law of XT,1 , say µ, admits a
unique solution, then the set of all polynomials is dense in L2 (µ) (see Akhiezer
[1965, Theorem 2.3.3]) and hence also in L1 (µ). A sufficient conditions for the
uniqueness of a solution to this moment problem is
k Ex XT,1 ≤ C k!
Rk
(6.3)
for some constants C > 0 and R > 0 (see Reed and Simon [1975, Example
X.4]). This condition can be assured by the existence of exponential moments
of XT,1 around 0, that is, the moment generating function Ex [euXT,1 ] is finite
for all u ∈ (−ε, ε). Indeed, since u 7→ Ex [euXT,1 ] is real analytic on (−ε, ε),
there exist an open interval J ⊆ (−ε, ε) and constants C > 0 and R > 0 such
that for all u ∈ J
k
d Ex [euXT,1 ] ≤ C k!
k
du
Rk
6.2. Pricing - Variance Reduction
171
(see, e.g., Krantz and Parks [2002, Corollary 1.2.9]). Hence in particular (6.3)
is satisfied.
In order to present the positive impact of our variance reduction method
graphically, we implemented the following affine stochastic volatility model,
which was initially proposed by Bates [2000]. The price process is specified as
St = S0 eXt with dynamics
R
Xt
r − V2t − λVt R (eξ − 1)F (dξ)
dt
d
=
Vt
b − βVt
√
Vt
0
dB
dZ
t,1
t
p
√
√
+
+
,
dBt,2
0
σρ Vt σ 1 − ρ2 Vt
where B is a 2-dimensional Brownian motion and Z a pure jump process
in R with jump intensity λv and exponentially distributed jump sizes, i.e.,
ξ
F (ξ) = 1c e− c , for some parameter c ∈ R+ . Figure 6.1 illustrates the comparison between the Monte-Carlo simulation for European Call prices with and
without variance reduction. In this example we use the following parameters:
S0
V0
Strike
r
b
β
σ
ρ
λ
c
10
0.1
9
0.04
0.08
0.7
0.03
0
1.5
0.05
The polynomial which we take to approximate the payoff function is of degree
10 and is chosen such as to minimize the approximation error with respect
to the supremum norm in a certain interval (depending on the support of
the probability distribution). Concerning computation time, we remark that
beside the one-time calculation of the matrix exponential, the only additional
computational effort resulting from the use of the control variates is the evaluation of a polynomial in each loop. In our MATLAB code, this causes an
increase of computation time of less than 50% per replication. Observing that
one can achieve the same accuracy by using 100 times less replications through
the polynomial control variates, the computation time (in our MATLAB implementation) can be decreased by a factor of more than 65.
Let us finally remark that our variance reduction technique is of particular
interest for affine models for which the generalized Riccati ODEs (see Duffie
et al. [2003]) determining the characteristic function cannot be explicitly
solved. Moreover, it can also be applied to derivatives involving several assets,
provided that their dynamics are described by a polynomial process. This can
simply be done by approximating European payoff functions depending on
several variables with multivariate polynomials.
172
Chapter 6. Applications of Polynomial Processes
Figure 6.1: Comparison: Monte-Carlo simulation for European Call prices
with and without variance reduction.
Appendix A
Symmetric Cones and
Euclidean Jordan Algebras
A.1
Important Definitions
In this section (V, h·, ·i) denotes an Euclidean Jordan algebra and K the corresponding symmetric cone of squares, as introduced in Definition 3.1.3, Definition 3.1.1 and Theorem 3.1.6. The aim of this appendix is to review some
important notions and results related to Euclidean Jordan algebras.
A.1.1
Determinant, Trace and Inverse
We denote by R[λ] the polynomial ring over R in a single variable λ and, for
x ∈ V , we define R[x] := {p(x) | p ∈ R[λ]}. Then we have
R[x] = R[λ]/J (x)
with the ideal J (x) := {p ∈ R[λ] | p(x) = 0}. Since R[λ] is a principal
ring, J (x) is generated by a polynomial, which is referred to as the minimal
polynomial if the leading coefficient is 1. Its degree is denoted by m(x). We
have
m(x) = min k > 0 | e, x, x2 , . . . , xk are linearly dependent .
Furthermore, the rank of V is the number
r := max m(x),
x∈V
(A.1)
which is bounded by n = dim(V ). An element x is said to be regular if
m(x) = r. By Faraut and Korányi [1994, Proposition II.2.1], there exist
173
174
Appendix A. Symmetric Cones and Euclidean Jordan Algebras
unique polynomials a1 , . . . , ar on V such that the minimal polynomial of every
regular element x is given by
f (λ; x) = λr − a1 (x)λr−1 + a2 (x)λr−2 + · · · + (−1)r ar (x).
Using this fact, one introduces, for any x ∈ V , the determinant det(x) and
trace tr(x) as
det(x) := ar (x) and tr(x) := a1 (x).
Remark A.1.1. In order to distinguish between elements of V and linear
maps on V , we use the notations Tr(A) and Det(A) for A ∈ L(V ).
An element x is said to be invertible if there exists an element u ∈ R[x]
such that x ◦ y = e. Since R[x] is associative, y is unique. It is called the
inverse of x and is denoted by y = x−1 .
Remark A.1.2. We remark that the notions “rank”, “trace” and “determinant” are motivated by the fact that for the Euclidean Jordan algebra of real
r × r symmetric matrices, the rank is equal to r, and ar (x) and a1 (x) are the
usual determinant and trace, respectively. The inverse is also the usual one.
Note that the determinant is not multiplicative in general, but we have
the following.
Proposition A.1.3. For all z ∈ V and x, y ∈ R[z],
det(x ◦ y) = det(x) det(y).
Moreover, det(e) = 1 and tr(e) = r. In particular,
det(x) det(x−1 ) = det(x ◦ x−1 ) = 1.
A.1.2
Idempotents, Spectral and Peirce Decomposition
An element p of V is called idempotent if p2 = p, and two idempotents p, q are
called orthogonal if p ◦ q = 0. It is important to note that for idempotents this
notion of orthogonality coincides with the notion of orthogonality with respect
to the scalar product h·, ·i on V . Finally, a non-zero idempotent element is
called primitive if it cannot be expressed as a sum of non-zero orthogonal
idempotents.
The following results are cornerstones of the theory of Jordan algebras:
A.1. Important Definitions
175
Spectral Decomposition A set of mutually orthogonal primitive idempotents p1 , . . . , pr such that p1 + · · · + pr = e is called a Jordan frame of
rank r corresponding to the rank of the Euclidean Jordan algebra, as
defined in (A.1). The spectral decomposition theorem (see Faraut and
Korányi [1994, Theorem III.1.2]) states that, for every x ∈ V , there
exists a Jordan frame p1 , . . . , pr and real numbers λ1 , . . . , λr , such that
x=
r
X
λk pk ,
k=1
where the numbers λk are uniquely determined by x. Moreover, x is
an element of the symmetric cone K if and only if λk ≥ 0 for all
k ∈ {1, . . . , r}. Again motivated by symmetric matrices, λ1 , . . . , λr are
referred to as eigenvalues.
Peirce decomposition 1 Let c be an idempotent in V . Define, for k =
0, 1/2, 1 the subspaces V (c, k) := {x ∈ V : c ◦ x = kx}. Then, by Faraut
and Korányi [1994, Proposition IV.1.1], V can be written as the direct
orthogonal sum
V = V (c, 1) ⊕ V (c, 1/2) ⊕ V (c, 0).
Moreover, any x ∈ V decomposes with respect to this decomposition
into
x = x1 + x 1 + x0 ,
2
(A.2)
where xk ∈ V (c, k) for k = 0, 1/2, 1.
By Faraut and Korányi [1994, Proposition IV.1.1], the Peirce spaces
V (c, k) satisfy certain so called Peirce multiplication rules:
V (c, 1) ◦ V (c, 0) = {0},
(V (c, 1) + V (c, 0)) ◦ V (c, 1/2) ⊂ V (c, 1/2) ,
V (c, 1/2) ◦ V (c, 1/2) ⊂ V (c, 1) + V (c, 0),
P (V (c, 1/2)) V (c, 1) ⊂ V (c, 0).
(A.3)
(A.4)
(A.5)
(A.6)
Peirce decomposition 2 Let p1 , . . . , pr be a Jordan frame. Then V can be
written as the direct orthogonal sum
M
V =
Vij ,
(A.7)
i≤j
176
Appendix A. Symmetric Cones and Euclidean Jordan Algebras
where Vii = V (pi , 1) = Rpi and Vij = V (pi , 1/2) ∩ V (pj , 1/2) (see Faraut
and Korányi [1994, Theorem IV.2.1]).
Moreover, the projection onto Vii is given by the quadratic representation
P (pi ), and the projection onto Vij by 4L(pi )L(pj ).
If V is simple, then the dimension of Vij , i < j, denoted by d = dim Vij ,
is independent of i, j and the Jordan frame. It is called Peirce invariant.
If V is simple of dimension n and rank r, then we have by Faraut and
Korányi [1994, Corollary IV.2.6]
d
n = r + r(r − 1).
2
Corresponding to the decomposition (A.7), we can write for all x ∈ V
x=
r
X
i=1
xi +
X
xij ,
i<j
with xi ∈ Rpi and xij ∈ Vij . One can think of x as a symmetric r ×
r matrix, whose diagonal elements are the xi and whose off-diagonal
elements are the xij .
Example A.1.4. (i) Consider V = Rn equipped with the Euclidean scalar
product. Together with the Jordan product
x ◦ y = (x1 y1 , . . . , xn yn )
(element-wise multiplication in R), V is a Jordan algebra and the corresponding reducible cone is Rn+ . The idempotents are vectors consisting
only of zeros and ones. The non-zero primitive idempotents are the unit
vectors. The spectral decomposition (λ1 , . . . , λn ) are simply the coordinates of x in the Cartesian coordinate system.
(ii) Consider V = Sr , the space of real symmetric r × r-matrices. Here,
the idempotents correspond to the orthogonal projections and the nonzero primitive idempotents are the orthogonal projections on one-dimensional subspaces. In the spectral decomposition, λ1 , . . . , λr are the usual
eigenvalues of x and p1 , . . . , pr are the orthogonal projections on the
corresponding eigenvectors.
Concerning Peirce decomposition 1, the matrix of block form
Ik 0
0 0
A.1. Important Definitions
177
is an idempotent of V . The associated Peirce decomposition of a symmetric matrix is then given by
x1 x>
x1 0
0 x>
0 0
12
12
=
+
+
.
x12 x0
0 0
x12 0
0 x0
As already established above, Peirce decomposition 2 corresponds to




..
..
.
.




..



xij
.
r 
 X
X




.
.

xi
x=

+
..






.
i=1 
 i<j 
..
xij



.
..
.
.
.
Note that the dimension d of Vij , i < j, is 1 here.
A.1.3
Classification of Simple Euclidean Jordan Algebras
For sake of completeness, we here state the classification of all simple Euclidean
Jordan algebras and their corresponding irreducible cones. This classification
is summarized in the following table. Indeed, every simple Euclidean Jordan
algebra is isomorph to one of these cases.
Here, H and O denote the algebra of quaternions and octonions, respectively (see Faraut and Korányi [1994, page 84]). We denote by Herm(r, A) the
real vector space of Hermitian matrices with entries in A, where A corresponds
either to C, H or O.
K
V
Sr+
Sr
Herm+ (r, C)
Herm(r, C)
Herm+ (r, H)
dim V
rk V
d
r
1
r2
r
2
Herm(r, H)
r(2r − 1)
r
4
Lorentz cone
R × Rn−1
n
2
n−2
Exceptional cone
Herm(3, O)
27
3
8
1
r(r
2
+ 1)
178
A.2
Appendix A. Symmetric Cones and Euclidean Jordan Algebras
Some Results
In this section we collect a number of lemmas and propositions which are used
in the proofs of C 3. In most cases, we only cite the assertions without proofs,
as they can be found in Faraut and Korányi [1994]. For the sake of notational
convenience we always assume that V is a simple Euclidean Jordan algebra of
dimension n and rank r, equipped with the natural scalar product
h·, ·i : V × V → R,
hx, yi := tr(x ◦ y).
However, the particular form of the scalar product and the assumption that
V is simple is not always needed.
Lemma A.2.1. (i) Let a, b be idempotents in V . Then ha, bi ≥ 0. Moreover, ha, bi = 0 if and only if a ◦ b = 0.
(ii) Let a, b ∈ K and suppose that ha, bi = 0. Then a ◦ b = 0.
Proof. For (i) we follow the proof of Nomura [1993]: Let b = b0 + b 1 + b1
2
be the Peirce decomposition of b relative to a. Then ha, bi = ha ◦ b, bi =
1
kb1/2 k2 + kb1 k2 , from which the assertion follows.
2
For (ii) we follow the proof of Hertneck [1962]: Let a, b ∈ K and suppose
that ha, bi = 0. Since b ∈ K there exists c ∈ V such that b = c2 . Thus
0 = ha, bi = ha, c2 i = hL(a)c, ci.
But L(a) is positive semidefinite on V , and we conclude that L(a)c = a◦c = 0.
Hence
hb, a ◦ bi = hc2 , a ◦ c2 i = hc4 , ai = hc3 , a ◦ ci = 0,
and (ii) follows.
Proposition A.2.2. The following assertions hold true:
(i) An element x is invertible if and only if P (x) is invertible. Then we
have
P (x)x−1 = x,
P (x)−1 = P (x−1 ).
(ii) If x and y are invertible, then P (x)y is invertible and
(P (x)y)−1 = P (x−1 )y −1 .
A.2. Some Results
179
(iii) The differential map x 7→ x−1 is −P (x)−1 , that is,
d −1
(x + tu)|t=0 = −P (x−1 )u.
dt
(iv) det(P (x)y) = (det x)2 det y.
(v) ∇ ln det x = x−1 .
Proof. See Faraut and Korányi [1994, Proposition II.3.1, II.3.3 (i), II.3.3 (ii),
III.4.2 (i), III.4.2 (ii)]
Finally, let us introduce the following notation. For x and y in V we write
x y = L(x ◦ y) + [L(x), L(y)],
where [L(x), L(y)] = L(x)L(y) − L(y)L(x). The following transformation is
used to prove Lemma 3.4.6:
Definition A.2.3. Let c be an idempotent in V . For z in V (c, 12 ) we define
the Frobenius transformation τ (z) by τ (z) = exp(2 z c).
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