PROGRESSIVE ENLARGEMENTS OF FILTRATIONS
AND SEMIMARTINGALE DECOMPOSITIONS
Libo Li and Marek Rutkowski∗
School of Mathematics and Statistics
University of Sydney
NSW 2006, Australia
July 10, 2011
Abstract
We deal with decompositions of F-martingales with respect to the progressively enlarged
filtration G and we extend there several related results from the existing literature. Results on
G-semimartingale decompositions of F-local martingales are crucial for applications in financial
mathematics, especially in the context of modeling credit risk, where τ represents the moment
of occurrence of some credit event (e.g., the default event).
∗ The research of L. Li and M. Rutkowski was supported under Australian Research Council’s Discovery Projects
funding scheme (DP0881460).
1
2
Progressive Enlargements and Semimartingale Decompositions
Contents
1 Introduction
1.1
3
Applications to Financial Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Preliminaries
4
6
2.1
Conditional Distribution of a Random Time . . . . . . . . . . . . . . . . . . . . . . .
6
2.2
Multiplicative Construction of a Random Time . . . . . . . . . . . . . . . . . . . . .
6
2.3
Properties of Conditional Distributions . . . . . . . . . . . . . . . . . . . . . . . . . .
8
2.3.1
Hypotheses (H) and (HP ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
2.3.2
Complete Separability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2.3.3
Separability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2.3.4
Extended Density Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
3 Enlargements of Filtrations
10
3.1
Case of an Honest Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
3.2
General Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
3.3
Conditional Expectations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
3.3.1
Conditional Expectations under Hypothesis (HP ) . . . . . . . . . . . . . . . .
13
3.3.2
Conditional Expectations under the Extended Density Hypothesis . . . . . .
16
Properties of G-Local Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
3.4.1
G-Local Martingales under Hypothesis (HP ) . . . . . . . . . . . . . . . . . .
16
3.4.2
G-Local Martingales under the Extended Density Hypothesis . . . . . . . . .
19
Compensator of H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
3.5.1
Compensator of H under Complete Separability . . . . . . . . . . . . . . . .
20
3.5.2
Compensator of H under the Extended Density Hypothesis . . . . . . . . . .
20
3.4
3.5
4 Semimartingale Decompositions
22
4.1
Case of Continuous Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
4.2
Case of Discontinuous Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
4.2.1
Semimartingale Decomposition under Hypothesis (HP ) . . . . . . . . . . . .
24
4.2.2
Semimartingale Decomposition under the Extended Density Hypothesis . . .
27
4.2.3
Case of Multiplicative Construction . . . . . . . . . . . . . . . . . . . . . . .
28
5 Applications to Financial Mathematics
5.1
5.2
28
Martingale Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
5.1.1
Hypothesis (H) in a General Set-up . . . . . . . . . . . . . . . . . . . . . . .
30
5.1.2
Martingale Measures via Hypotheses (H) and (HP ) . . . . . . . . . . . . . .
31
Information Theory
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
L. Li and M. Rutkowski
1
3
Introduction
We work on a probability space (Ω, G, P) endowed with a filtration F satisfying the usual conditions
and we assume that τ is any random time with values in R̄+ defined on this space.
Definition 1.1. By an enlargement of F associated with τ (or, briefly, an enlargement of F) we
mean any filtration K = (Kt )t∈R+ in (Ω, G, P), satisfying the usual conditions, and such that: (i) the
inclusion F ⊂ K holds, meaning that Ft ⊂ Kt for all t ∈ R+ , and (ii) τ is a K-stopping time.
The following two particular enlargements of F were extensively studied in the literature (see, e.g.,
Dellacherie and Meyer [12], Jeulin [24, 25, 26], Jeulin and Yor [27, 28], and Yor [35, 36]; Nikeghbali
[31] (see Section 8 therein) provides a short overview of the most pertinent results, but mostly
under the assumptions that all F-martingales are continuous and/or the random time τ avoids all
F-stopping times.
Definition 1.2. The initial enlargement of F is the filtration G∗ = (Gt∗ )t∈R+ given by the equality
Gt∗ = ∩s>t (σ(τ ) ∨ Fs ) for all t ∈ R+ .
The initial enlargement does not seem to be well suited for the analysis of a random time since it
postulates that σ(τ ) ⊂ G0∗ , meaning that all the information about τ is already available at time 0. It
appears that the following notion of the progressive enlargement is more suitable for formulating and
solving problems associated with an additional information conveyed by observations of occurrence
of a random time τ .
Definition 1.3. The progressive enlargement of F is the minimal enlargement, that is, the smallest
filtration G = (Gt )t∈R+ , satisfying the usual conditions, such that F ⊂ G and τ is a G-stopping time.
More explicitly, Gt = ∩s>t Gso where we denote Gto = σ(τ ∧ t) ∨ Ft for all t ∈ R+ .
Let H be the filtration generated by the process Ht = 1{τ ≤t} . It is clear that F ∨ H ⊂ G ⊂ G∗ .
In fact, the inclusion F ∨ H ⊂ K necessarily holds for any enlargement of F. In what follows,
we will mainly work with the progressive enlargement G. However, we would like first to clarify
the relationships between various filtrations encountered in the existing literature. Note that some
definitions are only applicable to the case of an honest time (see Definition 3.1).
We are now in the position to describe very briefly the related results from the existing literature.
For any two filtrations F ⊂ K on a probability space (Ω, G, P), we say that the hypothesis (H 0 ) holds
for F and K under P if any (P, F)-semimartingale is also a (P, K)-semimartingale. By the classic
result due to Jacod [18], the hypothesis (H 0 ) is satisfied in the case of the initial enlargement of F
provided that τ is an initial time. Recall that τ is called an initial time if there exists a measure η
on (R̄+ , B(R̄+ )) such that the F-conditional distributions of τ are absolutely continuous with respect
to η, that is, Fdu,t η(du). This property is commonly referred to as the density hypothesis.
The hypothesis (H 0 ) should be contrasted with the stronger hypothesis (H) for F and K under P,
also known as the immersion property, which stipulates that any (P, F)-martingale is also a (P, K)martingale (see Brémaud and Yor [7]). In the case of the progressive enlargement G of a filtration
F through a random time τ defined on (Ω, G, P), the abovementioned hypothesis (H) for F and G
under P is well known to be equivalent to the hypothesis (H) introduced in Definition 2.4 (see,
for instance, Elliott et al. [15]) and thus no confusion may arise. However, the main hypothesis
examined in the present work are the hypothesis (HP ) and extended density hypothesis introduced
in Definition 2.4 and 2.7 respectively. the hypothesis (HP ) is clearly weaker than the hypothesis
(H) and it is satisfied, in particular, when a random time is constructed using the multiplicative
approach. Also, under mild technical assumption, it is equivalent to the (either complete or partial)
separability of the F-conditional distribution of τ (see Definitions 2.5 and 2.6).
On the other hand, the extended density hypothesis is a reformulation/extension of the concept
of density hypothesis: there exists an F-adapted increasing process D such that Fdu,t dDu (see
Definition 2.7). We do so to avoid the technical assumption on strict positivity of F-conditional
4
Progressive Enlargements and Semimartingale Decompositions
distribution of the random time τ . It is worth to point out that most results obtained for initial
times can be extended to this new setting without any serious difficulty.
The problem of the semimartingale decomposition of a (P, F)-martingale with respect to a progressive enlargement G was examined in several papers during the last thirty years. In particular,
the following properties are well known (see Yor [35] and Dellacherie and Meyer [12]):
(a) a (P, F)-martingale may fail to be a semimartingale with respect to a progressive enlargement G,
(b) any (P, F)-semimartingale stopped at τ is a (P, G)-semimartingale,
(c) any (P, F)-semimartingale is a (P, G)-semimartingale if τ is an honest time.
Having the above points in mind, if a process U is a (P, F)-local martingale. It is natural to ask
whether U is a (P, G)-semimartingale and to compute its (canonical) semimartingale decomposition
with respect to G. In the path-breaking paper by Jeulin and Yor [27] (see also Barlow [3, 4] and
Jeulin and Yor [28]), the authors derived the (P, G)-semimartingale decomposition of the stopped
process Uτ ∧t for any random time τ , as well as the (P, G)-semimartingale decomposition of U under
an additional assumption that τ is an honest time. The latter result was recently extended to the
case of initial times in papers by Nikeghbali and Yor [32], Jeanblanc and Le Cam [20] and El Karoui
et al. [15]. Of course, we are not in a position to discuss these results in detail here, although some
of them will be stated in what follows.
The paper is organized as follows.
In Section 2, we outline briefly the multiplicative approach to constructing of a random time
with a predetermined Azéma supermartingale that was developed in Li and Rutkowski [29]. we also
examine in this section some properties of F-conditional distributions of a random time under (HP)
hypothesis and extended density hypothesis.
In Section 3, we revisit and clarify the relationship between filtrations in the existing literature,
while paying special attention to the progressive enlargement of F with a random time. We provide
an alternative characterization of the filtration G. This alternative characterization is used in the
subsequent subsections in computations of conditional expectation of G-adapted processes under
hypothesis (HP ) and extended density hypothesis. With these results, we are then able to give
sufficient conditions for a class of G-adapted processes to a (P, G)-martingale.
In Section 4, we compute the semimartingale decompositions of (P, F)-martingales with respect
to the progressively enlarged filtration G and extend there several related results from the existing
literature. It should be stressed that is not assumed here that the pair (τ, P) was obtained using
the multiplicative system approach summarized in Section 2. Results on (P, G)-semimartingale
decompositions of F-local martingales are crucial for applications in financial mathematics, especially
modeling of credit risk where τ represents the moment of occurrence of some credit event (e.g., a
default).
For all unexplained concepts from the general theory of stochastic processes we refer to, for
instance, the monographs by Dellacherie [11] and Jacod and Shiryaev [19], as well as the survey
paper by Nikeghbali [31].
1.1
Applications to Financial Mathematics
In view of financial mathematics and applications, we shall provide two examples. As usual, one is
given a random time τ on a probability space (Ω, G, P) endowed with a filtration F satisfying the
usual conditions.
In view of model construction, it is natural to demand that any given model should be arbitrage
free. In the recent work of Coculescu et al. [9], the existence of an equivalent probability measure
for which the hypothesis (H) holds was shown to be a sufficient condition for a model with enlarged
filtration to be arbitrage-free, provided that the model based on the filtration F is arbitrage free.
In subsection 5.1, we sketch the most pertinent results from [9] and then show that under mild
technical assumptions, if (τ, P) satisfies the hypothesis (HP ) then there exists an equivalent probability measure for which hypothesis (H) holds. That is, if the market trading with information F is
5
L. Li and M. Rutkowski
arbitrage-free, then so is the market that trades with information described by G.
Next, we consider utility maximisation under asymmetric information. We give a brief sketch
on how the results obtain on G-semimartingale decomposition in Proposition 4.2 can be applied to
insiders information problems studied in the existing literature.
Given a (P, F)-semimartingale S, one argues as follows. Heuristically, different investors makes
different investment decisions based on the information they have, which is modeled by different
filtrations. The filtration F can be interpreted as information that is publicly available to all investors.
An investor is said to be an insider, if he has access to a larger filtration K that contains the reference
filtration F.
In particular, We refer to [1, 16] for studies on asymmetric information in financial markets and
[8, 5, 13] on anticipative approach to the insider problem. Broadly speaking, given a filtration K ⊃ F,
let K be the set of K-adapted x-superadmissible trading strategies, that is any trading strategies such
that the investors wealth process Vt (x) is strictly positive for all t ≥ 0. In the case of F-adapted
x-superadmissible strategies, it is described by the following condition
Z
Vt (x) := x +
φs dSs > 0,
∀t ≥ 0.
(0,t]
where x ≥ 0 is the initial wealth.
The anticipative or the forward integral approach (under certain technical assumptions) allows
one to write down an investor’s wealth process or hedging portfolio as a forward integral of their
K-adapted x-superadmissible trading strategy φ against the F-adapted process S, even if the process
S is not a K-semimartingale.
The mathematical formulation of the utility maximisation problem of insiders are given in [8].
We shall introduce it below and hint on how our results can be applied. First in [8] the author aims
to maximise insider utility and the optimization problem is formulated as follows,
!!
Z
UK (x) := max EP
φ∈K
U
φt d− St
x+
(1)
(0,T ]
where the integral is understood as a forward stochastic integrals and the function U is any utility
functions. In the case where the hypothesis (H 0 ) holds between F and K, the forward stochastic
integral can be replaced by a standard stochastic integral. With the results developed in this paper,
one can work with the filtration K = G been the progressive enlargement of the reference filtration
F with a positive random variable τ satisfying the extended density hypothesis. Although by doing
so, we loss the generality of the anticipative approach, as the process S may not even have to be a
semimartingale in the larger filtration K, but what we hope to do is to avoid technical assumptions
of Malliavin calculus and by keeping computations to F and G adapted processes only, one may be
able to provide more explicit computations.
Ankirchner and Imkeller have shown in [1] that if F and K are both finite utility filtrations for the
process S (see Definition 5.1) then the difference in expected logarithmic utility between investors
working with filtrations F and K where F ⊂ K depends only on the information drift of K relative to
F (see Definition 5.2). Therefore, as an application, we give in Section 5.2 under extended density
hypothesis and certain integrability conditions, an explicit representations for:
(i) the solution to optimization problem (1) with U (x) = log(x) and K = G
(ii) the information drift of G with respect to F.
We show that the progressive enlargement G of a finite utility filtration F for the process S with a
random time is once again a finite utility filtration for the process S.
6
Progressive Enlargements and Semimartingale Decompositions
2
Preliminaries
In this section, we deal with the most pertinent properties of F-conditional distributions of a random
time. First, we summarize in Section 2.2 the multiplicative approach to a random time with a given
in advance Azéma supermartingale G, as developed in Li and Rutkowski [29]. The first step is to
produce an F-conditional distribution of τ consistent with G using the concept of the multiplicative
system. Next, one constructs a random time with a predetermined F-conditional distribution on
a suitable extension of the underlying probability space. The construction provided in Section 2.2
motivates us to study in Section 2.3, various properties of F-conditional distributions that will play
an essential role in the study of enlarged filtrations.
2.1
Conditional Distribution of a Random Time
For an arbitrary random time τ , defined on some filtered probability space (Ω, F, F, P), we define
the (P, F)-submartingale F τ and the (P, F)-supermartingale Gτ = 1 − F τ by setting
Ftτ = P ( τ ≤ t | Ft ) ,
Gτt = P ( τ > t | Ft ) .
Note that the processes F τ and Gτ are positive and bounded by 1.
τ
Definition 2.1. The F-conditional distribution of τ is the random field (Fu,t
)u,t∈R̄+ given by
τ
Fu,t
= P ( τ ≤ u | Ft ) ,
∀ u, t ∈ R̄+ .
The F-conditional survival distribution of τ is the random field (Gτu,t )u,t∈R̄+ given by
τ
Gτu,t = P ( τ > u | Ft ) = 1 − Fu,t
,
∀ u, t ∈ R̄+ .
More generally, it is convenient to introduce also the following definition.
Definition 2.2. A random field (Fu,t )u,t∈R̄+ on a filtered probability space (Ω, F, F, P) is said to
be an F-conditional distribution of a random time if it satisfies:
(i) for every u ∈ R̄+ and t ∈ R̄+ , we have 0 ≤ Fu,t ≤ 1, P-a.s.,
(ii) for every u ∈ R̄+ , the process (Fu,t )t∈R̄+ is a (P, F)-martingale,
(iii) for every t ∈ R̄+ , the process (Fu,t )u∈R̄+ is right-continuous, increasing and F∞,t = 1.
A random field (Gu,t )u,t∈R̄+ is said to be an F-conditional survival distribution of a random time
whenever Fu,t = 1 − Gu,t is an F-conditional distribution of a random time.
The following result from Li and Rutkowski [29] shows that if a random field Fu,t satisfies
e P)
b
e F,
e F,
Definition 2.2 then there exists a random time τ on some extended probability space (Ω,
τ
such that Fu,t
= Fu,t .
Lemma 2.1. For any given F-conditional distribution Fu,t , there exists a random time τ on an
b ( τ ≤ u | Ft ) = Fu,t for all u, t ∈ R̄+ . Moreover,
extension of the usual probability space such that P
b |F = P |F holds for all t ∈ R+ .
the equality P
t
t
Usually, a random field Fu,t is not given in advance, but one rather deals with a predetermined
submartingale F . Hence the following assumption.
Assumption 2.1. We are given a submartingale F = (Ft )t∈R̄+ , defined on an underlying filtered
probability space (Ω, F, F, P), and such that 0 ≤ Ft ≤ 1 for every t ∈ R+ and F∞ = 1.
2.2
Multiplicative Construction of a Random Time
For a given in advance (P, F)-submartingale F , Li and Rutkowski [29] provided also a method of
constructing a random field Fu,t consistent with F , in the sense that Ft = Ft,t . We will now describe
briefly the method proposed in [29]. We start by recalling the concept of a multiplicative system
associated with a submartingale (see Meyer [30]).
7
L. Li and M. Rutkowski
Definition 2.3. Given a positive submartingale F = (Ft )t∈R̄+ , we say that (Cu,t )u,t∈R̄+ is a multiplicative system associated with F if the following conditions are satisfied:
(i) if u ≤ s ≤ t then Cu,s Cs,t = Cu,t and Cu,t = 1 if u ≥ t,
(ii) for any fixed u ∈ R+ , the process (Cu,t )t∈R̄+ is a predictable and decreasing,
(iii) for any fixed t ∈ R+ , the process (Cu,t )u∈R̄+ is a right-continuous and increasing,
(iv) the following equality holds, for all t ∈ R̄+ ,
EP ( Ct,∞ F∞ | Ft ) = Ft .
(2)
For the proof of the next result, see Corollary 2.1 in Li and Rutkowski [29].
Lemma 2.2. Let (Cu,t )u,t∈R̄+ be a multiplicative system associated with F . Then:
(i) for any fixed u, the process (Cu−,t Ft )t∈[u,∞] is a (P, F)-martingale,
(ii) the following relationship holds, for every 0 ≤ u < t,
Cu,t p Ft = Cu,t− Ft− .
(3)
The next lemma, borrowed from Li and Rutkowski [29], shows that for any positive submartingale
F satisfying Assumption 2.1, one can construct an F-conditional distribution Fu,t such that Ft,t = Ft .
Lemma 2.3. Let (Ft )t∈R̄+ be a submartingale satisfying Assumption 2.1. We define the random
field (Fbu,t )u,t∈R̄+ by setting
(
EP ( Fu | Ft ) , t ∈ [0, u),
(4)
Fbu,t =
Cu,t Ft ,
t ∈ [u, ∞],
where Cu,t is any multiplicative system associated with F . Then Fbu,t is an F-conditional distribution
and Fbt,t = Ft .
We are in the position to state the main result from Li and Rutkowski [29]. It is worth noting
that in order to establish Theorem 2.1, it suffices to combine Lemmas 2.1 and 2.3.
Theorem 2.1. Assume that we are given a submartingale (Ft )t∈R̄+ satisfying Assumption 2.1. Then
b P)
b of the filtered probability space (Ω, F, F, P)
b F,
b F,
there exists a random time τ on the extension (Ω,
b
such that the F-conditional distribution of τ under P equals
(
b ( τ ≤ u | Ft ) =
Fbu,t = P
EP ( Fu | Ft ) ,
Cu,t Ft ,
t ∈ [0, u],
t ∈ (u, ∞].
(5)
b ( τ > t | Ft ) = Gt is valid for every t ∈ [0, ∞]. Moreover, the equality
In particular, the equality P
b |F = P |F is satisfied for all t ∈ R+ .
P
t
t
Although the multiplicative system associated with the submartingale F may not be unique (see
Meyer [30]), the F-conditional survival distribution constructed from any multiplicative systems Cu,t
associated with F is always unique in this framework. Let us also observe that in Theorem 2.1 we
only deal with the random field (Cu,t )u∈R+ , t≥u .
Corollary 2.1. The random time τ constructed in Theorem 2.1 is with probability 1:
(i) a finite random time if and only if the process F is continuous at infinity, that is, the equality
F∞ = F∞− holds.
(ii) a strictly positive random time if and only if the process F is continuous at time zero, that is,
F0 = F0− = 0.
Under the following stronger assumption the uniqueness of the multiplicative system associated
with F holds and an explicit representation is known.
8
Progressive Enlargements and Semimartingale Decompositions
Assumption 2.2. The submartingale F = (Ft )t∈R̄+ defined on a filtered probability space (Ω, F, F, P)
is such that F∞ = 1 and the inequalities 0 < Ft ≤ 1 and 0 < Ft− ≤ 1 are valid for every t ∈ R+ .
Let the process B be given by B = A − G0 , where G = G0 + M − A is the Doob-Meyer
decomposition of the bounded (P, F)-supermartingale G, so that also G = M − B. Let us observe
that B is the F-predictable, increasing process that generates the supermartingale 1 − F . It is also
known that A is the (P, F)-dual predictable projection of the (non-F-adapted) increasing process
Ht = 1{τ ≤t} .
Proposition 2.1. Assume that F satisfies Assumption 2.2. Then the unique multiplicative system
associated with F satisfies, for every 0 ≤ u ≤ t,
R
Et − (0,·] (p Fs )−1 dBs
C0,t
R
=
(6)
Cu,t =
C0,u
E −
(p F )−1 dB
u
(0,·]
s
s
where Et (U ) stands for the stochastic exponential of the process U , that is, the unique solution to the
stochastic differential equation
dEt (U ) = Et− (U ) (p Ft )−1 dBt
with E0 (U ) = 1.
The following corollary provides sufficient conditions for the strict positivity of the random field
Fbu,t for all u, t > 0 (this property will also be referred to as the non-degeneracy of Fbu,t ).
b ( τ ≤ u | Ft ) given by (5). Under
Corollary 2.2. Consider the F-conditional distribution Fbu,t = P
b
Assumption 2.2, the random field Fu,t is strictly positive, except perhaps for Fb0,0 = F0 .
2.3
Properties of Conditional Distributions
We will now examine further properties of F-conditional distributions of τ . It shall become evident
that the (P, G)-semimartingale decomposition of (P, F)-martingales hinges on the properties of the
conditional distribution of τ .
2.3.1
Hypotheses (H) and (HP )
we recall the classic hypothesis (H) (see, for instance, Brémaud and Yor [7] or Elliott et al. [14])
and the weaker hypothesis (HP ) that was introduced in papers by Jeanblanc and Song [21] and Li
and Rutkowski [29].
Definition 2.4. A pair (τ, P) is said to satisfy the hypothesis (H) whenever for all 0 ≤ u ≤ s ≤ t
Fu,s = Fu,t .
(7)
A pair (τ, P) is said to satisfy the hypothesis (HP ) whenever for all 0 ≤ u ≤ s ≤ t
Fu,s Fs,t = Fs,s Fu,t .
(8)
The hypotheses of Definition 2.4 can be seen either as properties of a pair (τ, P) or, equivalently,
as properties of the associated F-conditional distribution Fu,t . It is clear that (H) implies (HP ). It
b is constructed as in Theorem 2.1 then the hypothesis (HP ) holds
is also known that if the pair (τ, P)
(see Proposition 3.1 in Li and Rutkowski [29]). By contrast, the hypothesis (H) is not necessarily
b obtained in Theorem 2.1.
satisfied by the pair (τ, P)
9
L. Li and M. Rutkowski
2.3.2
Complete Separability
Definition 2.5. We say that an F-conditional distribution Fu,t is completely separable if there exists
a positive (P, F)-martingale X and a positive, F-adapted, increasing process Y such that Fu,t = Yu Xt
for every u, t ∈ R+ such that u ≤ t.
It is easily seen that the complete separability of Fu,t implies that the hypothesis (HP ) holds.
Indeed, we have that Fu,s Fs,t = (Yu Xs )(Ys Xt ) = (Ys Xs )(Yu Xt ) = Fs,s Fu,t for all 0 ≤ u < s < t.
Recall that in the case of the multiplicative approach to an admissible construction of a random
time τ the F-conditional distribution Fbu,t is unique and satisfies the equality Fbu,t = Cu,t Ft = Qu,t
for all 0 ≤ u ≤ t where, for any fixed u ∈ R+ , the process (Qu,t )t≥u is a uniformly integrable
(P, F)-martingale. Using Corollary 2.1, we thus obtain the following result.
b ≤ u | Ft ) be given by formula (5).
Proposition 2.2. Let the F-conditional distribution Fbu,t = P(τ
Under Assumption 2.2, we have that, for all 0 ≤ u ≤ t,
R
Ft Et − (0,·] (p Fs )−1 dBs
R
= Yu Xt
Fbu,t =
Eu − (0,·] (p Fs )−1 dBs
where the strictly positive, F-adapted, increasing process Y is given by
Z
Yu = Eu −
(p Fs )−1 dBs
−1
(0,·]
and the strictly positive (P, F)-martingale X equals
Z
p
−1
Xt = Ft Et −
( Fs ) dBs .
(9)
(0,·]
Hence the random field Fbu,t is completely separable.
2.3.3
Separability
It appears that the property of complete separability is too restrictive, since it does not cover all
interesting cases. For this reason, we introduce also the weaker condition of separability that, in fact,
will be shown to be equivalent to the hypothesis (HP ) if a random time is strictly positive.
Definition 2.6. We say that an F-conditional distribution Fu,t is separable at v ≥ 0 if there exist
a positive (P, F)-martingale X v = (Xtv , t ∈ R+ ) and a positive, F-adapted, increasing process
Y v = (Yuv , u ∈ [v, ∞)) such that the equality Fu,t = Yuv Xtv holds for every v ≤ u ≤ t. Also, we say
that an F-conditional distribution Fu,t is separable if it is separable at all v > 0.
It is clear from Definitions 2.5 and 2.6 that an F-conditional distribution Fu,t is completely
separable whenever it is separable at 0.
Lemma 2.4. The following implications hold for any F-conditional distribution Fu,t :
(i) if Fu,t is separable at v ≥ 0 then it is also separable at all s ≥ v,
(ii) if Fu,t is separable at v ≥ 0 then the proportionality property (8) holds for all v ≤ u < s < t.
It appears that the separability of Fu,t implies the hypothesis (HP ) when F0 = 0, that is, when
the random time τ is strictly positive.
Proposition 2.3. If the F-conditional distribution of τ is separable and F0 = 0 then the hypothesis
(HP ) holds.
10
Progressive Enlargements and Semimartingale Decompositions
It seems natural to conjecture that in the non-degenerate case when Fu,t > 0 for all u, t > 0 the
converse implication also holds, that is, if the hypothesis (HP ) is valid then Fu,t is separable. This
is indeed the case, as the following result shows.
Proposition 2.4. Suppose that a non-degenerate F-conditional distribution Fu,t satisfies the hypothesis (HP ). Then the random field Fu,t is separable.
2.3.4
Extended Density Hypothesis
Definition 2.7. A pair of (τ, P) is said to satisfy the extended density hypothesis if there exists a
family of positive random field ms,t , which are (P, F)-martingales for t ≥ s ≥ 0 and an F-adapted,
increasing process D such that the F-conditional distribution Fu,t admits the following representation
Z
Fu,t =
ms,t dDs ,
u ≤ t.
[0,u]
The above definition can be seem as a natural extension to the density hypothesis. The results
obtained under extended density hypothesis are similar to those under density hypothesis. Nevertheless, it is introduced as it allows us to circumvent an awkward non-degeneracy condition of the
F-conditional distribution of a random time, which is needed in the proof of Lemma 3.5.
Remark 2.1. If the F-conditional distribution of τ is constructed through multiplicative systems
approach then the F-conditional distribution Fbu,t has the following integral representation
Z
Z
Z
Z
Ft Cs,t
Ft Cs,t
Ft Cs,t
Ft Cs,t
dB
=
F
−
dB
+
dB
=
dBs
Fbu,t = Ft −
s
t
s
s
pF
pF
pF
pF
s
s
s
s
[Lt ,t]
[Lt ,u]
[Lt ,u]
(u,t]
where the Ft -measurable random time Lt equals
Lt = sup 0 ≤ u ≤ t : Cu,t = 0 = sup 0 ≤ u ≤ t : Cu−,t = 0 .
(10)
If the Azéma supermartingale is strictly positive, then Lt = 0 for all t ≥ 0. This feature of
the multiplicative construction can be seen as an alternative motivation for the extended density
hypothesis.
3
Enlargements of Filtrations
We start by analyzing the basic properties of various enlargements of F associated with a random
time τ . Recall that the notion of an enlargement of F was introduced in Definition 1.1.
3.1
Case of an Honest Time
We examine first the case of an honest time, which was extensively studied in the literature (see,
e.g., Jeulin [24, 25, 26] and Jeulin and Yor [27, 28]).
Definition 3.1. A positive random variable τ defined on a filtered probability space (Ω, G, F, P) is
called an honest time if, for every t > 0, there exists an Ft -measurable random variable τt such that
τ is equal to τt on the event {τ ≤ t}, that is, τ 1{τ ≤t} = τt 1{τ ≤t} .
The following properties of honest times are well known (see, in particular, remark on page 76
in Dellacherie and Meyer [12]):
(a) an honest time τ is an F∞ -measurable random variable,
(b) the event {τ ≤ t} may be replaced by the event {τ < t} in Definition 3.1,
(c) a finite honest time can equivalently be defined as the end of an F-optional set.
To check that an honest time satisfies the hypothesis (HP ), it suffices to use the following result.
11
L. Li and M. Rutkowski
Lemma 3.1. [Yor [35]] The following properties are equivalent:
(i) τ is an honest time,
(ii) for every u < s, there exists an event Aus ∈ Fs such that {τ ≤ u} = Aus ∩ {τ ≤ s}.
Corollary 3.1. An honest time τ satisfies the hypothesis (HP ).
Proof. Let 0 ≤ u < s < t. Using Lemma 3.1, we obtain
Fu,s Fs,t = P(τ ≤ u | Fs ) P(τ ≤ s | Ft ) = P(Aus ∩ {τ ≤ s} | Fs ) P(τ ≤ s | Ft )
= 1Aus P(τ ≤ s | Fs ) P(τ ≤ s | Ft ) = P(τ ≤ s | Fs ) P(Aus ∩ {τ ≤ s} | Ft )
= P(τ ≤ s | Fs ) P(τ ≤ u | Ft ) = Fs,s Fu,t ,
which shows that equality (8) is satisfied, so that the hypothesis (HP ) holds.
The following families of σ-fields were introduced and examined in the context of honest times.
e = (Get )t∈R , G0 = (G 0 )t∈R and G
e 0 = (Ge0 )t∈R of σ-fields
Definition 3.2. We define the families G
t
t
+
+
+
by setting, for all t ∈ R+ ,
et ∈ Ft such that A ∩ {τ > t} = A
et ∩ {τ > t}},
Get = {A ∈ G | ∃ A
bt ∈ Ft such that A ∩ {τ ≤ t} = A
bt ∩ {τ ≤ t} ,
Gt0 = A ∈ G | ∃ A
et , A
bt ∈ Ft such that A = (A
et ∩ {τ > t}) ∪ (A
bt ∩ {τ ≤ t}) = Get ∩ Gt0 .
Get0 = A ∈ G | ∃ A
e 0 is uniquely characterized by the following equalities, for all t ∈ R+ ,
Note that the family G
Get0 ∩ {τ > t} = Ft ∩ {τ > t},
Get0 ∩ {τ ≤ t} = Ft ∩ {τ ≤ t},
(11)
e 0 does not depend on the choice of the σ-field G (it only depends on
meaning, in particular, that G
F and τ ).
e G0 and G
e 0 are well known (see Yor [35]):
The following properties of families G,
e
(a) for any random time τ , the family G is a filtration satisfying the usual conditions,
e 0 are not necessarily increasing,
(b) the families G0 and G
0
(c) if the family G is increasing then τ is an honest time,
e 0 is a filtration satisfying the usual conditions if and only if τ is an honest time.
(d) the family G
e and G ⊂ G
e 0 are valid and thus G
e is an enlargement
It is also clear that the inclusions G ⊂ G
0
e
of F for any random time τ , whereas G is an enlargement of F only when τ is an honest time.
e is convenient when dealing with the decomposition of F-martingales
The choice of the filtration G
with respect to enlarged filtration when we only examine processes stopped at τ (not necessarily an
honest time). In fact, one can use for this purpose any enlargement K that is admissible before τ , in
e (for any
the sense of Definition 3.3 (see, e.g., Guo and Zhang [17]). It is apparent the filtrations G
0
e
random time) and G (for an honest time) are admissible before τ .
Definition 3.3. An enlargement K of a filtration F is said to be admissible before τ if the equality
Kt ∩ {τ > t} = Ft ∩ {τ > t} holds for every t ∈ R+ .
e is no longer
When one deals with processes that are not stopped at τ then the choice of G
e
suitable, since the information conveyed by G after τ is too large, in general. The smaller filtration
e 0 can be used (note that G
e 0 ⊂ G)
e when τ is an honest time, but, obviously, it cannot be applied
G
e 0 coincides in fact with
to a general random time. As we shall see in what follows, the filtration G
the progressive enlargement G when τ is an honest time (see Jeulin [24] or part (ii) in Lemma 3.2).
e 0 was widely adopted as the natural choice of an enlargement of F in
This feature explains why G
the literature dealing with an honest time.
12
Progressive Enlargements and Semimartingale Decompositions
3.2
General Case
In the case of a general random time, we find it convenient to introduce the following notion, which
e 0.
hinges on a natural modification of the definition of the family G
b = (Gbt )t∈R is defined by setting, for every t ≥ 0,
Definition 3.4. The family G
+
et ∈ Ft and A
bτ,t ∈ Gt∗ such that A = (A
et ∩ {τ > t}) ∪ (A
bτ,t ∩ {τ ≤ t})}.
Gbt = {A ∈ G | ∃ A
We note that, for all t ∈ R+ ,
Gbt ∩ {τ > t} = Ft ∩ {τ > t},
Gbt ∩ {τ ≤ t} = Gt∗ ∩ {τ ≤ t}.
(12)
It is easily seen that the σ-field Gbt is uniquely characterized by conditions (12). The next result
b coincides in fact with the progressive enlargement G.
shows that the family G
b
Lemma 3.2. (i) If τ is any random time then G = G.
0
e = G.
b
(ii) If τ is an honest time then G = G
Proof. Recall that Gt = ∩s>t (σ(τ ∧ s) ∨ Fs ) and Gt∗ = ∩s>t (σ(τ ) ∨ Fs ) (see Definitions 1.2 and 1.3).
To show that Gbt = Gt , it suffices to check that conditions (12) are satisfied by Gt . The following
relationship for all t ∈ R+ is immediate,
Ft ∩ {τ > t} ⊂ Gt ∩ {τ > t} ⊂ Get ∩ {τ > t} = Ft ∩ {τ > t}.
This shows Gt ∩ {τ > t} = Ft ∩ {τ > t}, while on the other hand,
Gt ∩ {τ ≤ t} = ∩s>t (σ(τ ∧ s) ∨ Fs ) ∩ {τ ≤ t} = ∩s>t (σ(τ ) ∨ Fs ) ∩ {τ ≤ t} = Gt∗ ∩ {τ ≤ t}
since σ(τ ∧ s) ∩ {τ ≤ t} = σ(τ ) ∩ {τ ≤ t} for every s > t. For part (ii), we start by noting that the
e0 ⊂ G
b are obviously satisfied. From part (i), we know that G = G
b for any random
inclusions G ⊂ G
e0 = G
b for an honest time.
time and thus G = G
b (and hence G) is admissible before τ . When dealing with a
It is easy to see that the filtration G
semimartingale decomposition of an F-martingale after τ we will use the following concept.
Definition 3.5. We say that an enlargement K is admissible after τ if the equality Kt ∩ {τ ≤ t} =
Gt∗ ∩ {τ ≤ t} holds for every t ∈ R+ .
b (and thus G) is admissible after τ for any random time. Note also
It is clear that the filtration G
that if an enlargement K is admissible before and after τ then simply K = G.
Lemma 3.3. For any integrable, G-measurable random variable X and any enlargement K = (Kt )t≥0
admissible after τ we have that, for any t ∈ R+ ,
EP 1{τ ≤t} X Kt = lim EP 1{τ ≤t} X σ(τ ) ∨ Fs .
(13)
s↓t
Proof. It suffices to show that
EP 1{τ ≤t} X Kt = EP 1{τ ≤t} X Gt∗ .
(14)
The second equality in (13) will then follow from Corollary 2.4 in [34] since Gt∗ = ∩s>t (σ(τ ) ∨ Fs ).
To establish (14), we will first check that, for every A ∈ Kt ,
EP 1A 1{τ ≤t} X = EP 1A 1{τ ≤t} EP ( X | Gt∗ ) .
13
L. Li and M. Rutkowski
Since, by assumption, Kt ∩ {τ ≤ t} = Gt∗ ∩ {τ ≤ t}, there exists an event B ∈ Gt∗ such that A ∩
{τ ≤ t} = B ∩ {τ ≤ t}. Consequently,
EP 1A 1{τ ≤t} X = EP 1B 1{τ ≤t} X = EP 1B 1{τ ≤t} EP (X | Gt∗ ) = EP 1A 1{τ ≤t} EP (X | Gt∗ ) .
Hence
EP (1{τ ≤t} X | Kt ) = EP 1{τ ≤t} EP (X | Gt∗ ) | Kt = 1{τ ≤t} EP (X | Gt∗ ),
since the random variable 1{τ ≤t} EP ( X | Gt∗ ) is Kt -measurable.
3.3
Conditional Expectations
bT : R̄+ × Ω → R and we use the notation (u, ω) 7→ U
bu,T (ω).
For a fixed T > 0, we consider the map U
b
We postulate that UT is a B(R̄+ ) ⊗ FT -measurable map, so that Uτ,T is a σ(τ ) ∨ FT -measurable
random variable. We will need the following auxiliary lemma.
Lemma 3.4. The following inclusions hold for every t ∈ R+ :
(i) {τ > t} ⊂ {Gt > 0}, P-a.s.,
(ii) {τ ≤ t} ⊂ {Ft > 0}, P-a.s.
Proof. Let us denote A = {Ft = 1} = {P(τ ≤ t | Ft ) = 1}. Since A ∈ Ft
Z
Z
Z
P(A) =
Ft dP =
P(τ ≤ t | Ft ) dP =
1{τ ≤t} dP = P(A ∩ {τ ≤ t}).
A
A
A
Hence A = {Ft = 1} = {Gt = 0} ⊂ {τ ≤ t}, P-a.s., and thus {τ > t} ⊂ {Gt > 0}, P-a.s.
For part (ii), let us denote B = {Gt = 1} = {P(τ > t | Ft ) = 1}. Since B ∈ Ft
Z
Z
Z
P(B) =
Gt dP =
P(τ > t | Ft ) dP =
1{τ >t} dP = P(B ∩ {τ > t}).
B
B
B
Hence B = {Gt = 1} = {Ft = 0} ⊂ {τ > t}, P-a.s., and thus {τ ≤ t} ⊂ {Ft > 0}, P-a.s.
Remark 3.1. Part (i) in Lemma 3.4 can also be demonstrated as follows. Let τ0 = inf {t ∈ R+ :
Gt = 0 or Gt− = 0}. Since G is a supermartingale, it is equal to zero after τ0 and thus
P(τ0 < τ ) = EP (1{τ0 <τ } ) = EP (Gτ0 1{τ0 <∞} ) = 0.
This in turn implies that {τ > t} ⊂ {Gt > 0}, P-a.s.
Remark 3.2. Let us set, by convention, 0/0 = 0. Then, in view Lemma 3.4, the quantities
1{τ >t} G−1
and 1{τ ≤t} Ft−1 are well defined, P-a.s.
t
In the rest of the paper, we shall formulate results assuming that the F-conditional distribution
of the given in advance random time τ , satisfies either (i) hypothesis (HP) or (ii) the extended density
hypothesis, which are introduced in Definitions 2.4 and 2.7 respectively. In addition, the special case
of complete separability will also be mentioned.
3.3.1
Conditional Expectations under Hypothesis (HP )
The following lemma, which extends formula (20) to any t ∈ [0, T ], corresponds to Theorem 3.1 in
El Karoui et al. [15] where the case of an initial time was studied. Let us recall that the concept
of an initial time was introduced in the study of the hypothesis (H 0 ) under the initial enlargement
of F by some random variable (see Jacod [18] and Section 4 below). However, since it formally
corresponds to the co-called density hypothesis (see Jacod [18] and Hypothesis 2.1 in El Karoui et
al. [15]), the concept of an initial time can be studied in the context of any (that is, not necessarily
initial) enlargement of F.
14
Progressive Enlargements and Semimartingale Decompositions
Lemma 3.5. Let U·,T : R̄+ × Ω → R be a B(R̄+ ) ⊗ FT -measurable map. Assume that the pair (τ, P)
is such that the hypothesis (HP ) holds and the random variable Uτ,T is P-integrable.
(i) For every t ∈ [0, T ), we have that
et,T + 1{τ ≤t} U
bτ,t,T
EP (Uτ,T | Gt ) = 1{τ >t} U
(15)
where
et,T = (Gt )−1 EP 1{τ >t} Uτ,T | Ft = (Gt )−1 EP
U
Z
(t,∞]
Uu,T dFu,T Ft
(16)
and, for all 0 ≤ u ≤ t < T ,
bu,t,T = (1 − Gt )−1 EP (Ft,T Uu,T | Ft ).
U
(17)
(ii) If, in addition, Fu,t is completely separable so that Fu,t = Yu Xt for u ≤ T then (16) yields
Z
EP 1{T ≥τ >t} Uτ,T | Ft = EP XT
(t,T ]
Uu,T dYu Ft
(18)
and (17) becomes
bu,t,T = (Xt )−1 EP (XT Uu,T | Ft ).
U
(19)
Proof. The derivation of (16) is rather standard. Note that the hypothesis (HP ) is not needed here
and we may take t ∈ [0, T ]. It suffices to take Uu,T = g(u)1A for a Borel measurable map g : R+ → R
and an event A ∈ FT such that the random variable Uτ,T = g(τ )1A is P-integrable. Using part (i)
in Lemma 3.4 and the classic formula for conditional expectation with respect to Gt , we obtain, on
the event {τ > t},
EP 1{τ >t} Uτ,T | Ft
= (Gt )−1 EP 1A EP 1{τ >t} g(τ ) | FT Ft
EP (1{τ >t} Uτ,T | Gt ) =
P(τ > t | Ft )
Z
Z
= (Gt )−1 EP 1A
g(u) dFu,T Ft = (Gt )−1 EP
g(u)1A dFu,T Ft
(t,∞]
= (Gt )−1 EP
(t,∞]
Z
Uu,T dFu,T
(t,∞]
Ft ,
which yields (16). Let us take any t ∈ [0, T ). To establish (17), we need to compute EP ( Uτ,T | Gt )
on the event {τ ≤ t}. By an application of Lemma 3.3,
EP 1{τ ≤t} Uτ,T Gt = lim EP 1{τ ≤t} Uτ,T σ(τ ) ∨ Fs .
s↓t
We first compute for 0 ≤ t < s < T , the conditional expectation EP 1{τ ≤t} Uτ,T σ(τ ) ∨ Fs . Recall
that the hypothesis (HP ) means that the equality Fu,s Fs,T = Fs,s Fu,T holds for all 0 ≤ u < s < T ,
which implies that Fs,T dFu,s = Fs,s dFu,T for any fixed t < s < T and u ∈ [0, t].
Hence for any bounded, σ(τ ) ∨ Fs -measurable random variable Hτ,s we obtain
EP 1{τ ≤t} Hτ,s Uτ,T = EP EP 1{τ ≤t} Hτ,s Uτ,T | FT = EP
Z
1{u≤t} Hu,s Uu,T dFu,T
[0,∞]
Z
= EP
Hu,s (Fs,s )−1 Fs,T Uu,T dFu,s
[0,t]
bu,s,T dFu,s
Hu,s U
[0,t]
Z
= EP
Hu,s (Fs,s )−1 EP (Fs,T Uu,T | Fs ) dFu,s
[0,t]
Z
= EP
bτ,s,T
= EP 1{τ ≤t} Hτ,s U
15
L. Li and M. Rutkowski
since {τ ≤ t} ⊂ {τ ≤ s} ⊂ {Fs,s > 0}, P-a.s. (see part (ii) in Lemma 3.4). This in turn yields
bτ,s,T
EP 1{τ ≤t} Uτ,T Gt = lim EP 1{τ ≤t} Uτ,T σ(τ ) ∨ Fs = lim 1{τ ≤t} U
s↓t
s↓t
= lim 1{τ ≤t} (1 − Gs )
s↓t
−1
EP ( Fs,T Uu,T | Fs )u=τ
bτ,t,T
= 1{τ ≤t} (1 − Gt )−1 EP ( Ft,T Uu,T | Ft )u=τ = 1{τ ≤t} U
where the penultimate equality holds by right continuity of the filtration F together with the right
continuity of the processes Gu and Fu,t in u. This completes the proof of formula (17) and thus
equality (15) is established for t ∈ [0, T ]. For part (ii), we observe that if, in addition, the random
field Fu,t is completely separable then formulae (18) and (19) follow easily from equations (16) and
(17), respectively.
Remark 3.3. Let us observe that for t = 0, we first obtain on the event {τ = 0}
bτ,s,T = EP 1{τ =0} H0,s U
b0,s,T
EP 1{τ =0} Hτ,s Uτ,T = EP 1{τ =0} Hτ,s U
where, on the event {τ = 0} ⊂ {τ ≤ s} ⊂ {Fs > 0},
b0,s,T = (Fs )−1 EP (Fs,T U0,T | Fs ).
U
In the second step, we get, on the event {τ = 0} ⊂ {F0 > 0},
b0,0,T
b0,s,T = 1{τ =0} U
EP 1{τ =0} Uτ,T G0 = lim 1{τ =0} U
s↓0
where (see (17))
b0,0,T = (F0 )−1 EP (F0,T U0,T | F0 ) = EP (P(τ = 0 | FT )U0,T | F0 )
U
P(τ = 0 | F0 )
where F0,T = P(τ = 0 | FT ). We conclude that
Z
1
EP
Uu,T dP(τ ≤ u | FT ) F0
P(τ > 0 | F0 )
(0,∞]
Z
1
EP
Uu,T dP(τ ≤ u | FT ) F0 .
+ 1{τ =0}
P(τ = 0 | F0 )
[0]
EP (Uτ,T | G0 ) = 1{τ >0}
Remark 3.4. For t = T , we have that
eT,T + 1{τ ≤T } Uτ,T ,
EP (Uτ,T | GT ) = 1{τ >T } U
(20)
where formula (16) in Lemma 3.5 yields
Z
EP 1{τ >T } Uτ,T FT
e
= (GT )−1
Uu,T dFu,T .
UT,T =
P(τ > T | FT )
(T,∞]
(21)
Remark 3.5. Assume that the pair (τ, P) is such that the hypothesis (H) holds. Then formula (17)
simplifies as follows
bu,t,T = (1 − Gt )−1 EP (Ft,T Uu,T | Ft ) = (Ft )−1 EP (Ft,t Uu,T | Ft ) = EP (Uu,T | Ft ).
U
bτ,T = g(τ ), we have that U
bu,t,T = g(u) and
In particular, if U
et,T = (Gt )−1 EP 1{τ >t} g(τ ) | Ft = (Gt )−1 EP
U
Z
(t,∞]
g(u) dFu Ft .
16
3.3.2
Progressive Enlargements and Semimartingale Decompositions
Conditional Expectations under the Extended Density Hypothesis
Under the extended density hypothesis, Lemma 3.5 can be reformulated as follows.
Lemma 3.6. Let U·,T : R̄+ × Ω → R be a B(R̄+ ) ⊗ FT -measurable map. Let Assumption 2.7 be
satisfied and the random variable Uτ,T be P-integrable.
(i) For every t ∈ [0, T ), we have that
et,T + 1{τ ≤t} U
bτ,t,T
EP (Uτ,T | Gt ) = 1{τ >t} U
where
et,T = (Gt )−1 EP 1{τ >t} Uτ,T | Ft = (Gt )−1 EP
U
Z
Uu,T dFu,T
(t,∞]
(22)
Ft
(23)
and, for all 0 ≤ u ≤ t < T ,
bu,t,T = (mu,t )−1 EP (mu,T Uu,T | Ft ).
U
(24)
Proof. One only needs to modify to proof of Lemma 3.5 on the event {τ ≤ t}. For any bounded,
σ(τ ) ∨ Fs -measurable random variable Hτ,s we obtain
Z
EP 1{τ ≤t} Hτ,s Uτ,T = EP EP 1{τ ≤t} Hτ,s Uτ,T | FT = EP
1{u≤t} Hu,s Uu,T dFu,T
[0,∞]
Z
= EP
Hu,s mu,T Uu,T dDu
Z
= EP
Hu,s (mu,s )
[0,t]
EP (mu,T Uu,T | Fs )mu,s dDu
[0,t]
Z
= EP
−1
bu,s,T dFu,T
Hu,s U
bτ,s,T
= EP 1{τ ≤t} Hτ,s U
[0,t]
and by taking limit
bτ,T
lim EP 1{τ ≤t} U
s↓t
Ft = lim(mu,s )−1 EP (mu,T Uu,T | Fs )u=τ
s↓t
= (mu,t )−1 EP (mu,T Uu,T | Ft )u=τ .
This concludes the proof.
3.4
Properties of G-Local Martingales
b : R̄+ × R+ × Ω → R and we use the notation (u, t, ω) 7→ U
bu,t (ω). We
We consider the map U
b
say that U is an F-optional map if it is B(R̄+ ) ⊗ O(F)-measurable where O(F) is the F-optional
b·,t is B(R̄+ ) ⊗ Ft -measurable and the process (U
bt,t )t∈R
σ-field in R+ × Ω. In that case, the map U
+
is F-optional, in the usual sense. We will sometimes need an additional assumption that the process
bt,t )t∈R is F-predictable.
(U
+
3.4.1
G-Local Martingales under Hypothesis (HP )
We consider an arbitrary random time τ associated with G, in the sense that G is the Azéma
supermartingale of τ . The following result is an important step towards a (P, G)-semimartingale
decomposition of a (P, F)-local martingale, which will be established in the subsequent section.
Proposition 3.1. Assume that the hypothesis (HP ) holds and 0 < Fu,t ≤ 1 for every 0 < u ≤ t.
Let Ū be a P-integrable process such that
et + 1{τ ≤t} U
bτ,t ,
Ūt = 1{τ >t} U
(25)
17
L. Li and M. Rutkowski
e is an F-adapted, P-integrable process and U
b is an F-optional map such that the random
where U
b
bt,t )t∈R is F-predictable. Let the
variable Uτ,t is P-integrable for all t ∈ R+ and the process (U
+
following conditions hold:
(i) the process (Wt )t≥0 is a (P, F)-local martingale where
Z
et Gt +
bu,u dFu ,
Wt = U
U
(26)
(0,t]
(ii) for every u > 0 and s > 0, the process (Fs,t Uu,t )t≥u∨s is a (P, F)-local martingale.
Then the process (Ūt )t≥0 is a (P, G)-local martingale.
Proof. We merely sketch the proof since it proceeds along the similar lines as the proofs of Propositions 5.1 and 5.6 in El Karoui et al. [15]. In particular, we skip the details regarding suitable
localization and measurability arguments. In view of the following decomposition,
et 1{τ >t} + U
bτ,τ 1{τ ≤t} + (U
bτ,t − U
bτ,τ )1{τ ≤t} .
Ūt = Ūt 1{τ >t} + Ūτ 1{τ ≤t} + (Ūt − Ūτ )1{τ ≤t} = U
It is sufficient to examine the following two subcases corresponding to conditions (i) and (ii) respectively:
(i) the case of a process Ū stopped at τ ,
(ii) the case of a process Ū starting at τ with Ūτ = 0.
Case (i). Let us first assume that a G-adapted process Ū is stopped at τ so that
bτ ,
et + 1{τ ≤t} U
Ūt = 1{τ >t} U
(27)
et )t∈R is an F-adapted process and (U
bt := U
bt,t )t∈R is an F-predictable process. We start
where (U
+
+
by noting that, for every 0 ≤ s < t,
bτ | Gs )
bτ | Gs ) + EP (1{τ ≤s} U
bτ | Gs ) = EP (1{s<τ ≤t} U
EP (1{τ ≤t} U
Z
bu dHu Fs + 1{τ ≤s} U
bτ
= 1{τ >s} (Gs )−1 EP
U
(s,t]
= 1{τ >s} (Gs )−1 EP
Z
(s,t]
= 1{τ >s} (Gs )
−1
Z
EP
(s,t]
bu dAu Fs + 1{τ ≤s} U
bτ
U
b
bτ
Uu dFu Fs + 1{τ ≤s} U
since the dual (P, F)-predictable projection of H satisfies H p = A and Ft = F0 − Mt + At . Hence,
for every 0 ≤ s < t,
Z
−1
−1
e
b
bτ
EP (Ūt | Gs ) = 1{τ >s} (Gs ) EP (Ut Gt | Fs ) + 1{τ >s} (Gs ) EP
Uu dFu Fs + 1{τ ≤s} U
(s,t]
−1
−1
= 1{τ >s} (Gs ) EP (Wt − Ws | Fs ) + 1{τ >s} (Gs )
es + 1{τ ≤s} U
bτ = Ūs
= 1{τ >s} U
es Gs | Fs ) + 1{τ ≤s} U
bτ
EP (U
where we used the assumption that the process W given by (26) is a (P, F)-martingale.
bτ,t where
Case (ii). We now consider a G-adapted process Ū starting at τ so that Ūt = 1{τ ≤t} U
bt,t = 0. We need to show that the equality EP (1{τ ≤t} U
bτ,t | Gs ) = 1{τ ≤s} U
bτ,s holds for every
U
0 ≤ s < t. From part (i) in Lemma 3.5, we obtain, for every 0 ≤ s < t,
Z
bτ,t | Gs ) = 1{τ >s} (Gs )−1 EP
bu,t dFu,t Fs
EP (1{τ ≤t} U
U
(s,t]
+ 1{τ ≤s} (1 − Gs )−1 EP (Fs,t Uu,t | Fs )|u=τ = I1 + I2 .
18
Progressive Enlargements and Semimartingale Decompositions
We first assume that s > 0. Recall that we assume that the hypothesis (HP ) holds and 0 < Fu,t ≤ 1
−1
for every 0 < u ≤ t. Hence, for 0 < s ≤ u ≤ t, we can write dFu,t = Fs,t d(Fu,u Fs,u
) where the
s
−1
process Du := Fu,u Fs,u is increasing and F-adapted. Consequently,
Z
bu,t | Fu ) dDus Fs
I1 = 1{τ >s} (Gs )−1 EP
EP (Fs,t U
(s,t]
= 1{τ >s} (Gs )
−1
Z
EP
(s,t]
s b
Fs,u Uu,u dDu Fs = 0
bu,u = 0.
where we first used assumption (ii) and next the equality U
+
−
It remains to examine the case s = 0. We denote Uτ,t
= max (Uτ,t , 0) and Uτ,t
= max (−Uτ,t , 0).
Therefore, for all t > 0,
Z
bτ,t Ft F0
bu,t dFu,t F0 = 1{τ >0} (G0 )−1 EP lim EP 1{s≤τ ≤t} U
I1 = 1{τ >0} (G0 )−1 EP
U
s↓0
(0,t]
= 1{τ >0} (G0 )−1 EP lim EP
s↓0
+ − b
b
1{s≤τ ≤t} Uτ,t Ft F0 − EP lim EP 1{s≤τ ≤t} Uτ,t Ft F0 .
↓0
By the monotone convergence theorem for conditional expectations, we obtain
Z
bu,t dFu,t F0
I1 = 1{τ >0} (G0 )−1 lim EP
U
s↓0
(s,t]
= 1{τ >0} (G0 )−1 lim EP
Z
s↓0
(s,t]
bu,t | Fu ) dDs F0 = 0
EP (Fs,t U
u
bu,u = 0. Furthermore, using again assumption
where we used again assumption (ii) and the equality U
(ii), we obtain
bu,t | Fs )|u=τ = 1{τ ≤s} (1 − Gs )−1 Fs,s (U
bu,s )|u=τ = 1{τ ≤s} U
bτ,s .
I2 = 1{τ ≤s} (1 − Gs )−1 EP (Fs,t U
bτ,t is a (P, G)-martingale for t ≥ 0 and this completes the proof of the
We conclude that 1{τ ≤t} U
proposition.
The following corollary to Proposition 3.1 deals with the special case when the process U given by
bt,t )t∈R
(25) is continuous at τ . It is clear that under the assumptions of Corollary 3.2 the process (U
+
is F-predictable.
Corollary 3.2. Under the assumptions of Proposition 3.1 we postulate, in addition, that the equality
et− = U
bt,t holds for every t ∈ R+ . Then the process Ū is continuous at τ and condition (i) in
U
Proposition 3.1 can be replaced by the following condition:
(i0 ) the process (Wt )t≥0 is a (P, F)-local martingale where
Z
et Gt +
eu− dFu .
Wt = U
U
(28)
(0,t]
Assume that Fu,t is completely separable. Then the computations for case (ii) in the proof of
Proposition 3.1 can be simplified. Specifically, using part (ii) in Lemma 3.5, we obtain, for every
0 ≤ s < t,
Z
bτ,t | Gs ) = 1{τ >s} (Gs )−1 EP
bu,t dYu Fs + 1{τ ≤s} (Xs )−1 EP (Xt U
bu,t | Fs )|u=τ
EP (1{τ ≤t} U
Xt U
(s,t]
= 1{τ >s} (Gs )−1 EP
Z
(s,t]
−1
= 1{τ >s} (Gs )
Z
EP
(s,t]
EP (Wu,t | Fu ) dYu Fs + 1{τ ≤s} (Xs )−1 EP (Wu,t | Fs )|u=τ
bτ,s
Wu,u dYu Fs + 1{τ ≤s} (Xs )−1 (Wu,s )|u=τ = 1{τ ≤s} U
19
L. Li and M. Rutkowski
bu,u = 0 in
where we used condition (ii) in the penultimate equality and the equality Wu,u = Xu U
the last one. This leads to the following corollary in which the assumptions of Proposition 3.1 are
maintained.
Corollary 3.3. Assume that the F-conditional distribution of τ is completely separable, so that
Fu,t = Yu Xt where Y is a positive increasing process and X a strictly positive (P, F)-martingale.
Then condition (ii) in Proposition 3.1 can be replaced by the following condition:
bu,t )t≥u is a (P, F)-martingale.
(ii’) For every u ≥ 0, the process (Xt U
3.4.2
G-Local Martingales under the Extended Density Hypothesis
In Proposition 3.1, one has to assume the F-conditional distribution Fu,t is non-degenerate, since
the random measure Dus := Fu,u (Fs,u )−1 is not always well defined if the F-conditional distribution
is degenerate. Thus in order to remove such technicality, one can replace it with Assumption 2.7.
Proposition 3.2. Suppose the extended density hypothesis holds with density mu,t and F-adapted increasing process D. Then condition (ii) in Proposition 3.1 can be replaced by the following condition:
(ii*) for every u > 0, the process (mu,t Uu,t )t≥u is a (P, F)-local martingale.
Proof. The only adjustment needed is in case (ii). Let the process Ū start at τ so that Ūt =
bτ,t , where U
bt,t = 0. From Lemma 3.6,
1{τ ≤t} U
bτ,t | Gs ) = 1{τ >s} (Gs )−1 EP
EP (1{τ ≤t} U
Z
(s,t]
−1
+ 1{τ ≤s} (mu,s )
b
Uu,t dFu,t Fs
EP (mu,t Uu,t | Fs )|u=τ = I1 + I2 .
We again consider the integrals separably. The first integral I1 is given by
Z
Z
−1
−1
b
b
I1 = 1{τ >s} (Gs ) EP
Uu,t dFu,t Fs = 1{τ >s} (Gs ) EP
mu,t Uu,t dDu Fs
(s,t]
(s,t]
Z
bu,t | Fu dDu Fs = 0
= 1{τ >s} (Gs )−1 EP
EP mu,t U
(s,t]
bu,u = 0. The integral I2 simplifies to
where we first used assumption (ii) and next the equality U
bu,t | Fs )|u=τ = 1{τ ≤s} (mu,s )−1 mu,s (U
bu,s )|u=τ = 1{τ ≤s} U
bτ,s .
I2 = 1{τ ≤s} (mu,s )−1 EP (mu,t U
bτ,t is a (P, G)-martingale for t ≥ 0 and this completes the proof of the
We conclude that 1{τ ≤t} U
proposition.
3.5
Compensator of H
Our next goal is to compute the (P, G)-dual predictable projection of the process Ht = 1{τ ≤t} where
G is the progressive enlargement of F through τ . To this end, we will first compute the (P, F)-dual
predictable projection of H. Assume the Doob-Meyer decomposition of G is known and is given by
Gt = G0 + Mt − At .
It is well known that the (P, F)-dual predictable projection of H coincides with the F-predictable process of finite variation A. To find the (P, G)-dual predictable projection (that is, the G-compensator)
of H, it is enough to apply the following well known theorem, due to Jeulin and Yor [27] (see also
Guo and Zeng [17]).
20
Progressive Enlargements and Semimartingale Decompositions
Theorem 3.1 (Jeulin-Yor). Let τ be a random time with the associated Azéma supermartingale G.
Then the process
Z
1
Ht −
dAs
(0,t∧τ ] Gs−
is a (P, G)-martingale.
3.5.1
Compensator of H under Complete Separability
In this subsection, we work under the assumption that the F-conditional distribution of τ under P
is completely separable (see Definition 2.5), that is
P ( τ ≤ u | Ft ) = Yu Xt .
We assume, in addition, that the increasing process Y is (P, F)-predictable. It is worth noting that
both assumptions are satisfied in the multiplicative construction setting provided that Gt < 1 for
all t > 0.
By applying the integration by parts formula to F and using the assumption that Y is a predictable process, we obtain
Z
Z
Ft = Yt Xt = Y0 X0 +
Ys dXs +
Xs− dYs .
(0,t]
(0,t]
Hence, by the uniqueness of the Doob-Meyer decomposition, we conclude that dAs = Xs− dYs .
Therefore, from the Jeulin-Yor formula, we deduce that the process
Z
Xs−
dYs
Ht −
(0,τ ∧t] 1 − Xs− Ys−
is a (P, G)-martingale.
3.5.2
Compensator of H under the Extended Density Hypothesis
In this subsection, we examine the case where the random time τ satisfies the extended density
hypothesis (see Definition 2.7), that is
Z
P ( τ ≤ u | Ft ) =
ms,t dDs
[0,u]
Proposition 3.3. Assume that the extended density hypothesis holds and the increasing process D
is predictable. Then the (P, F)-dual predictable projection of H is given by
Z
p
Htp =
mu dDu .
(0,t]
Proof. We compute the Doob-Meyer decomposition of Ft = P ( τ ≤ t | Ft ). From the extended
density hypothesis, it is easy to see that
Z
Z
Ft,t = m0,t ∆D0 +
(mu,t − mu,u ) dDu +
mu dDu .
(0,t]
(0,t]
where mu := mu,u . The process D is predictable and increasing and thus
!p
Z
mu dDu
(0,t]
Z
=
(0,t]
p
mu dDu .
21
L. Li and M. Rutkowski
The Doob-Meyer decomposition can be computed in this case and is given by
Z
Z
Z
Ft,t = m0,t ∆D0 +
(mu,t − mu,u ) dDu +
(mu − p mu ) dDu +
= Mt1 + Mt2 +
(0,t]
Mt3 +
(0,t]
p
mu dDu
(0,t]
At
where the processes M i for i = 1, 2, 3 are
Z
Mt1 = m0,t ∆D0 ,
Mt2 =
(mu,t − mu,u ) dDu ,
Mt3 =
(0,t]
Z
(mu − p mu ) dDu .
(0,t]
It is enough to show Mt2 is a (P, F)-martingale, since M 1 and M 3 are clearly (P, F)-martingales. We
have, for all 0 ≤ s ≤ t,
!
Z
2
EP Mt Fs = EP
(mu,t − mu,u ) dDu Fs
(0,t]
!
!
Z
Z
= EP
(mu,t − mu,u ) dDu Fs + EP
(mu,t − mu,u ) dDu Fs
(0,s]
(s,t]
!
Z
Z
=
EP ( mu,t − mu,u | Fs ) dDu + EP
EP ( mu,t − mu,u | Fu ) dDu Fs
(0,s]
(s,t]
Z
=
(mu,s − mu,u ) dDu = Ms2 ,
(0,s]
so that M 2 is a (P, F)-martingale. The predictable increasing process is thus given by
p
R
(0,t]
mu dDu .
Remark 3.6. In the case of a random time τ constructed using the multiplicative system approach,
the compensator for H is simply given by the process A, that is the predictable process of finite
variation in the Doob-Meyer decomposition of G. This property is also a simple application of
Proposition 3.3. Let us recall first that in the multiplicative system approach, if Gt < 1 for all t ≥ 0,
then the F-conditional distribution satisfies the extended density hypothesis and
Z
Z
Ft Cs,t
Ft Cs,t
Fu,t =
dBs =
dBs .
(29)
pF
pF
s
s
[Lt ,u]
[0,u]
where Bt = At − G0 . The claim easily follows, since p mu = p
Fu
pF
u
=1
In what follows, we no longer assume the process D is predictable. We impose instead alternative
assumptions on the process mu := mu,u .
Proposition 3.4. Assume the process m is a special semimartingale with the canonical decomposition given by m = N + P . If the predictable covariation of N and D exists then the (P, F)-dual
predictable projection of H is given by the formula
Z
p
Htp = hN, Dit +
mu dDup
(30)
(0,t]
where p mu = Nu− + Pu .
Proof. We will compute the Doob-Meyer decomposition of Ft = P ( τ ≤ t | Ft ). Under the extended
density hypothesis,
Z
Z
Ft = m0,t ∆D0 +
(mu,t − mu,u ) dDu +
mu dDu .
(0,t]
(0,t]
22
Progressive Enlargements and Semimartingale Decompositions
By using the canonical decomposition of m, we obtain
Z
Z
Z
Ft = m0,t ∆D0 +
(mu,t − mu,u ) dDu −
(Nt − Nu ) dDu + Nt Dt +
Pu dDu
(0,t]
(0,t]
(0,t]
Z
= Mt1 + Mt2 − Mt3 + [N, D]t +
(Nu− + Pu ) dDu .
(0,t]
The second equality is obtained after we apply integration by parts formula to the product N D and
by setting
Z
Z
Z
1
2
3
Mt := m0,t ∆D0 , Mt :=
(mu,t − mu,u ) dDu , Mt :=
(Nt − Nu ) dDu +
Du− dNu .
(0,t]
(0,t]
(0,t]
It is easy to see that all above processes are (P, F)-martingales. Using the dual predictable projection
of D, we can write
Z
Z
Z
p
(Nu− + Pu ) dDu =
(Nu− + Pu ) d(Du − Du ) +
(Nu− + Pu ) Dup .
(0,t]
(0,t]
(0,t]
Finally, the predictable covariation of N and D exists and thus we can write
[N, D] = [N, D] − hN, Di + hN, Di .
It is thus easy to deduce from the above that the predictable increasing part in the Doob-Meyer
decomposition of F is given by
Z
hN, Dit +
(Nu− + Pu ) dDup .
(0,t]
To conclude the proof of equality (30), it suffices to observe that p mu = Nu− + Pu .
4
Semimartingale Decompositions
Our next goal is to show that if a pair (τ, P), given on some filtered probability space (Ω, F, P), satisfies
the hypothesis (HP ) then every (P, F)-semimartingale is also a (P, G)-semimartingale, where G is the
progressive enlargement of F by τ . To do so, we calculate the (P, G)-semimartingale decomposition
for a (P, F)-local martingale.
4.1
Case of Continuous Martingales
Throughout this subsection, we postulate that all (P, F)-martingales are continuous. This is fact the
standing assumption in several papers dealing with enlargements of filtrations. In particular, it was
postulated in the paper by Yor [36], which will be employed in this subsection. It is worth noting that
in that case the dual optional projection H o and the dual predictable projection H p with respect to
(P, F) coincide. Consequently, it will be enough to consider the Doob-Meyer decomposition of G.
As before, we consider a random time τ with the Azéma supermartingale G. Recall that we write
G = G0 + M − A to denote the Doob-Meyer decomposition of the bounded (P, F)-supermartingale
G. Under the present assumptions, we have that H o = H p = A. Under the complete separability
of Fu,t , the equality Gt = 1 − Yt Xt holds and thus dMt = −Yt dXt and dAt = Xt dYt .
For any bounded, Borel measurable function g : R+ → R, we define the following (P, F)martingales
Z ∞
g
g
Mt = EP (g(τ ) | Ft ), Nt = EP
g(u) dAu Ft .
0
Note that under the present assumption, the (P, F)-martingales M g and N g (which are also (P, F)martingales) are necessarily continuous. We will need two auxiliary lemmas. The proof of Lemma
4.1 is straightforward and thus it is omitted.
23
L. Li and M. Rutkowski
Lemma 4.1. The following equality holds
Mtg − Ntg = EP (g(τ )1{τ <t} | Ft ) −
t
Z
g(u) dAu .
0
Lemma 4.2. Let a pair (τ, P) be such that the associated F-conditional distribution Fu,t of τ under
P is completely separable, so that Fu,t = Yu Xt for u ∈ [0, t], where Y is a positive, F-adapted,
increasing process and X is a positive (P, F)-martingale. We have that
Z tZ s
g(u)γs dFu,s dXs
Mtg − Ntg =
0
0
where γs = (Xs )−1 .
Proof. Using Lemma 4.1, we obtain
Mtg
−
Ntg
Z
t
Z
t
Z
t
= EP (g(τ )1{τ <t} | Ft ) −
g(u) dAu =
g(u) dFu,t −
g(u) dAu
0
0
0
Z t
Z t
Z t
Z t
=
g(u)Xt dYu −
g(u)Xu dYu =
g(u)
dXs dYu
0
0
0
u
Z tZ s
Z tZ s
=
g(u) dYu dXs =
g(u)γs dFu,s dXs
0
0
0
0
since dAu = Xu dYu . This follows from the equality Gt = 1 − Xt Yt and the continuity of the
(P, F)-martingale X (recall that Y is an increasing process and X is also a (P, F)-martingale).
In view of Lemma 4.2, the following result is a consequence of Theorem 2 in Yor [36]. Note that
formulae (31) and (32) yield the canonical decomposition of a (P, G)-semimartingale U .
Proposition 4.1. (i) Assume that all (P, F)-martingales are continuous. Let U be a (P, F)-local
martingale such that
Z t
νu |dhU, Xiu | < ∞
0
where
Z
t
Z
|γt | dFu,t =
νt =
0
t
(Xt )−1 dFu,t = (Xt )−1 Ft,t .
0
Then the process Ū given by the equality
Z t∧τ
Z
Ūt = Ut −
(Gu )−1 dhU, M iu −
0
t∨τ
(Xu )−1 dhU, Xiu
is a (P, G)-local martingale.
(ii) Furthermore, the process Ū can be represented as follows
Z t∧τ
Z t∨τ
−1
Ūt = Ut −
(Gu ) dhU, M iu +
1{τ <u} (Fu )−1 dhU, M iu .
0
(31)
τ
(32)
τ
Proof. Part (i) is an immediate consequence of Lemma 4.2 and Theorem 2 in Yor [36]. To establish
part (ii), we observe that from part (i) we have that Ū = U − Ā, where the process Ā is given by
et 1{τ >t} + A
bτ,t 1{τ ≤t} ,
Āt = A
where in turn, for every t ∈ R+ ,
Z t
Z t
et =
A
(Gu )−1 dhU, M iu =
(Gu )−1 dhU, Giu
0
0
(33)
(34)
24
Progressive Enlargements and Semimartingale Decompositions
and, for every 0 ≤ u ≤ t,
Z
t
bu,t = A
eu− +
A
(Xs )
−1
Z
t
(Fs )−1 dhU, M is
eu− +
dhU, Xis = A
u
(35)
u
since dMs = −Ys dXs and Fs = Ys Xs .
By applying Theorem 2 in Yor [36] to an honest time, it is possible to show (see Example E.2 in
Yor [36]) that if U is a (P, F)-local martingale then the process
Z
Ūt = Ut −
t∧τ
(Gu )−1 dhU, M iu +
Z
0
t∨τ
(Fu )−1 dhU, M iu
τ
is a (P, G)-local martingale. This shows that Proposition 4.1 furnishes a natural extension of the
canonical decomposition of a G-semimartingale in the case of an honest time.
Remark 4.1. For an honest time, more general canonical decompositions of G-semimartingales were
obtained in papers by Jeulin and Yor in [27, 28] without the assumption that all (P, F)-martingales
are continuous. Let us briefly describe the most pertinent representation. Consider an honest time
τ defined on a probability space (Ω, G, P). Assume that U is a (P, F)-local martingale. Let B p be
the dual predictable projection of the process
Z
Bt =
∆Us dHs = ∆Uτ 1{τ ≤t}
(0,t]
e = hU, M i + B p . Then the process
and let C
Z
Z
−1 e
Ūt = Ut −
(Gu− ) dCu +
(0,t∧τ ]
eu
(Fu− )−1 dC
(36)
(τ,t∨τ ]
is a (P, G)-local martingale (see Theorem 2 in [27] or Theorem 15 in [28]).
4.2
Case of Discontinuous Martingales
We will now relax the assumption that all (P, F)-martingales are continuous.
4.2.1
Semimartingale Decomposition under Hypothesis (HP )
Proposition 4.2 furnishes a (P, G)-semimartingale decomposition of a (P, F)-local martingale U under
the hypothesis (HP ). On the one hand, this result can be seen as a counterpart of Proposition 5.9 in
El Karoui et al. [15] who dealt with the case of an initial time (see also Jeanblanc and Le Cam [20]
who examined both initial and honest times). In the special case of the multiplicative construction,
it is also related to Theorem 7.1 in Jeanblanc and Song [21].
Remark 4.2. if (τ, P) satisfies hypothesis (HP ), then for all 0 ≤ u ≤ s ≤ t,
Fu,s
Fs,t = Fu,t .
Fs,s
(37)
The inclusion {Fs,s = 0} ⊂ {Fu,s = 0} holds and by convention, 0/0 = 0.
Proposition 4.2. Let (τ, P) be any pair such that the hypothesis (HP ) holds and the F-conditional
distribution of τ satisfies 0 < Fu,t ≤ 1 for every 0 < u ≤ t. If U is a (P, F)-local martingale then the
process Ū is a (P, G)-local martingale where
Z
Z
−1
e
Ūt = Ut −
(Gu ) d[U, M ]u + (∆Aτ − ∆Uτ )1{τ ≤t} −
(Fτ,u )−1 d[U, Fτ,· ]u
(38)
(0,t∧τ ]
(τ,t∨τ ]
25
L. Li and M. Rutkowski
et is given by (39). Equivalently, Ū is given by
where A
Z
Z
Z
Ūt = Ut −
1{τ >u} (Gu )−1 d[U, M ]u −
∆Uu dHu −
(0,t]
[0,t]
1{τ <u} (Fτ,u )−1 d[U, Fτ,· ]u .
(0,t]
Proof. Although our set-up is more general than the one studied in El Karoui et al. [15], some
arguments used below are similar to those employed in the proof of Proposition 5.9 in [15]. We start
by noting that, by the optional stopping, we may and do assume, without loss of generality, that
U is a uniformly integrable (P, F)-martingale. We note that Ū = U − Ā, where Ā is a G-optional
e given by
process of finite variation given by (33) with the process A
Z
et =
A
(Gu )−1 d[U, M ]u
(39)
(0,t]
bu,t given by, for all 0 ≤ u ≤ t,
and with A
Z
bu,t = A
eu− + ∆Uu +
A
(Fu,s )−1 d[U, Fu,· ]s
(40)
(u,t]
where ∆Uu = Uu − Uu− . We need to show that the process
et + 1{τ ≤t} U
bτ,t = (Ut − A
et )1{τ >t} + (Ut − A
bτ,t )1{τ ≤t}
Ūt = 1{τ >t} U
(41)
et− = U
bt,t holds for every t ∈ R+ ,
is a (P, G)-local martingale. It is easy to check that the equality U
bt,t is F-predictable. We also observe that
so that the process Ū is continuous at τ and the process U
b
the map U is F-optional. In view of Proposition 3.1 and Corollary 3.2, it thus suffices to show that
the process
Z
et )Gt −
eu− ) dGu
(Ut − A
(Uu− − A
(42)
(0,t]
bu,t − U
bu,u ))t≥s∨u are (P, F)-local martingales. To this end,
and processes (Fs,t (U
et )Gt = Ut− dGt + Gt− dUt + d[U, G]t − A
et− dGt − Gt dA
et
d (Ut − A
and
et )Gt − (Ut− − A
et− ) dGt = Gt− dUt + d[U, G]t − Gt dA
et = Gt− dUt − d[U, B]t
d (Ut − A
where [U, B]t is a (P, F) martingale by Yœsurp’s Lemma. So that the process given by (42) is a
(P, F)-local martingale.
Let us now fix u > 0. We observe that for every t ≥ u
bu,t = Ut − A
bu,t = Ut − A
eu− − ∆Uu − Zu,t
U
where we denote
Z
Zu,t =
(Fu,v )−1 d[U, Fu,· ]v .
(u,t]
By applying the integration by parts formula, we obtain, for t ≥ u ∨ s,
bu,t ) = Ut− dFs,t + Fs,t− dUt + d[U, Fs,· ]t − Zu,t− dFs,t − Fs,t dZu,t .
d(Fs,t U
bu,t )t≥u∨s is a (P, F)-(local) martingale, it is sufficient to show that,
To show that the process (Fs,t U
for all t ≥ u ∨ s,
d[U, Fs,· ]t − Fs,t dZu,t = 0.
(43)
26
Progressive Enlargements and Semimartingale Decompositions
By Remark 4.2, for all u ≤ s ≤ t,
Fs,t dZu,t =
i
Fs,t
Fs,t h Fu,s
Fs,t Fu,s
d[U, Fu,· ]t =
d U,
Fs,· =
d[U, Fs,· ]t = d[U, Fs,· ]t .
Fu,t
Fu,t
Fs,s
Fu,t Fs,s
t
Similarly, for all s ≤ u ≤ t, we have
Fs,t dZu,t =
Fs,u
Fu,u Fu,t
= Fs,t and thus
h F
i
Fs,t
Fs,u Fu,t
Fs,u
s,u
d[U, Fu,· ]t =
d[U, Fu,· ]t =
d[U, Fu,· ]t = d U,
Fu,· = d[U, Fs,· ]t .
Fu,t
Fu,u Fu,t
Fu,u
Fu,u
t
This shows that (43) is indeed satisfied. Using Corollary 3.2, we conclude that Ū is a (P, G)-local
martingale. Finally, we observe that Ū can be represented as follows
et 1{τ >t} − (A
eτ − + ∆Uτ )1{τ ≤t} − Zτ,t 1{τ ≤t}
Ūt = Ut − A
et 1{τ >t} − A
eτ 1{τ ≤t} + (∆A
eτ − ∆Uτ )1{τ ≤t} − Zτ,t 1{τ ≤t}
= Ut − A
et∧τ + (∆A
eτ − ∆Uτ )1{τ ≤t} − Zτ,t∨τ ,
= Ut − A
which in turn leads to representation (38).
In the special case where the F-conditional distribution of τ is completely separable, we obtain
the following corollary to Proposition 4.2.
Corollary 4.1. Assume that the F-conditional distribution of τ is completely separable, that is, the
F-conditional distribution of τ is given by Fu,t = Yu Xt for every 0 ≤ u ≤ t. If U is a (P, F)-local
martingale then the process Ū is a (P, G)-local martingale where
Z
Z
eτ − ∆Uτ )1{τ ≤t} −
Ūt = Ut −
(Gu )−1 d[U, M ]u + (∆A
(Xu )−1 d[U, X]u
(0,t∧τ ]
(τ,t∨τ ]
e is given by (39).
where the process A
The next corollary shows that if τ satisfies the (HP ) hypothesis and non-degeneracy condition
then the (P, G)-semimartingale decomposition of (P, F)-martingales closely resembles the case where
τ is assumed to be a honest time.
Corollary 4.2. Under the assumptions of Proposition 4.2 and Cs,u := Fs,u (Fu )−1 is predictable in
u ≥ s for every s ≥ 0. If U is a (P, F)-martingale then the process Ū given by
Z
Z
eτ − ∆Uτ )1{τ ≤t} +
Ūt = Ut −
(Gu )−1 d[U, M ]u − (∆A
(Fu )−1 d[U, M ]u
(0,t∧τ ]
(τ,t∨τ ]
is a (P, G)-semimartingale. If one further assumes the (P, F)-martingale U has no jump at τ then
Z
Z
Ūt = Ut −
(Gu )−1 d[U, M ]u +
(Fu )−1 d[U, M ]u
(0,t∧τ ]
(τ,t∨τ ]
which closely resembles the close where τ is a honest time.
Proof. We first note that Cs,u is a multiplicative system associated with G = 1 − F . Integration by
parts formula gives
dFs,u = Cs,u dFu + Fu− dCs,u .
We only need to focus on the decomposition after τ .
Z
Z
Z
(Fs,u )−1 d[U, Fs,· ]u =
(Fu Cs,u )−1 Cs,u d[U, F ] +
(Fs,u )−1 Fu− d[U, Cs,· ]u s=τ
s=τ
(s,t∨s]
(s,t]
(s,t]
Z
Z
=
(Fu )−1 d[U, F ]u +
(Fu Cs,u )−1 Fu− d[U, Cs,· ]u ,
(s,t]
(s,t]
s=τ
27
L. Li and M. Rutkowski
the Doob-Meyer decomposition of F shows that
Z
Z
−1
=−
(Fs,u ) d[U, Fs,· ]u s=τ
(s,t∨s]
where
Ets :=
Z
(s,t∨s]
(Fu )−1 d[U, B]u +
(s,t∨s]
(Fu )−1 d[U, M ]u Z
s=τ
+ Ets s=τ
(Fu Cs,u )−1 Fu− d[U, Cs,· ]u .
(s,t∨s]
By assumption Gt = 1 − Ft is strictly positive for t > 0 and Corollary 2.2 in [29], tells us that the
multiplicative system Cu,s is unique and must satisfy the following stochastic differential equation
dCs,u = −Cs,u− (Fup )−1 dBu = −Cs,u (Fu− )−1 dBu ,
which implies Ets = 0.
4.2.2
Semimartingale Decomposition under the Extended Density Hypothesis
Recall that a pair (τ, P) is said to satisfy the extended density hypothesis if there exist a F-adapted
increasing process D and, for every u ≥ 0, a (P, F)-martingale (mu,t )t≥u such that
Z
Fu,t = P(τ ≤ u | Ft ) =
ms,t dDs
[0,u]
The G-semimartingale decomposition obtained under the density hypothesis can be seen as a special
case where the increasing process D is non-random.
Corollary 4.3. Under the extended density hypothesis, if U is a (P, F)-local martingale then the
process Ū is a (P, G)-local martingale where
Z
Z
eτ − ∆Uτ )1{τ ≤t} −
Ūt = Ut −
(Gu )−1 d[U, G]u + (∆A
(mτ,u )−1 d[U, mτ,· ]u
(44)
(0,t∧τ ]
(τ,t∨τ ]
e given by (39).
with the process A
Proof. In view of Lemma 3.2 and parts of the proof of Proposition 4.2, it suffices to show that
bu,t )t≥u is a martingale for t ≥ u where
(mu,t U
bu,t = Ut − A
eu− − ∆Uu − Zu,t
U
et given by (39) and
with A
Z
Zu,t =
(mu,v )−1 d[U, mu,· ]v .
(u,t]
To this end, we apply the integration by parts formula to obtain
bu,t ) = Ut− dms,t + ms,t− dUt + d[U, ms,· ]t − Zu,t− dms,t − mu,t dZu,t
d(mu,t U
= Ut− dms,t + ms,t− dUt − Zu,t− dmu,t
which is a (P, F)-martingale for t ≥ u.
If we assume, in addition, that ms,t = Xt aθ for some positive (P, F)-martingale X and some
positive process aθ then
Z
Z
Fu,t =
ms,t dDs = Xt
as dBs = Xt Yu
[0,u]
where Yu =
R
[0,u]
[0,u]
as dDs . In that case, the F-conditional distribution of τ is completely separable.
28
Progressive Enlargements and Semimartingale Decompositions
Corollary 4.4. Assume that the pair (τ, P) satisfies the extended density hypothesis and the Fconditional distribution is completely separable under P. Let U be a (P, F)-local martingale continuous
at τ . Then the process Ū is a (P, G)-local martingale, where
Z
Z
Ūt = Ut −
(Gu )−1 d[U, M ]u +
(ms,u )−1 d[U, ms,· ]u s=τ .
(0,t∧τ ]
(τ,t∨τ ]
Proof. It suffices to apply Corollary 4.1 and observe that, for every fixed u, the equality ms,u = as Xu
holds.
4.2.3
Case of Multiplicative Construction
The representation of the process Ū provided in Corollary 4.5 corresponds to formula (6) in Jeanblanc
and Song [21]. It can also be seen as a counterpart of the canonical semimartingale decomposition
obtained for honest times (see (36)).
b was constructed
Corollary 4.5. Assume that Gt < 1 and Gt− < 1 for every t > 0 and the pair (τ, P)
using the multiplicative approach of Section 2.2. Let U be a (P, F)-local martingale continuous at τ .
b G)-local martingale, where
Then the process Ū is a (P,
Z
Z
Ūt = Ut −
(Gu )−1 d[U, M ]u +
(Fu )−1 d[U, M ]u .
(0,t∧τ ]
(τ,t∨τ ]
Proof. It is easy to check that Assumption 2.2 holds. Recall also that X = F Eb where we denote
(see (9))
Z
Ebt = Et −
(p Fs )−1 dBs
(0,·]
where B is the predictable process that generates F so that dFt = dAt − dMt = dBt − dMt . Since
Eb is a predictable process of bounded variation, by applying Proposition 9.3.7.1 in Jeanblanc et al.
[23], we obtain
dXt = Ebt dFt + Ft− dEbt .
Consequently, we have that
bt
(Xt )−1 d[U, X]t = (Xt )−1 Ebt d[U, F ]t + (Xt )−1 Ft− d[U, E]
bt
= −(Ft )−1 d[U, M ]t + (Ft )−1 d[U, B]t + (Xt )−1 Ft− d[U, E]
Ft− Ebt−
d[U, B]t
= −(Ft )−1 d[U, M ]t + (Ft )−1 d[U, B]t − p
Ft Xt
where we used the fact that dEbt = Ebt− (p Ft )−1 dBt . Using the equality (see (3))
p
Ft Ebt = Ebt− Ft− ,
we obtain
(Xt )−1 d[U, X]t = −(Ft )−1 d[U, M ]t .
An application of Proposition 4.2 concludes the proof.
5
Applications to Financial Mathematics
In this final section, we sketch two applications of general results to particular problems arising in
the context of financial modeling.
29
L. Li and M. Rutkowski
5.1
Martingale Measures
In the recent paper by Coculescu et al. [9], the existence of an equivalent probability measure for
which the hypothesis (H) holds was shown to be a sufficient condition for a model with enlarged
filtration to be arbitrage-free, provided that the model based on the filtration F enjoys this property.
Proposition 5.1 shows that if the pair (τ, P) satisfies the hypothesis (HP ) (or, more precisely, the
complete separability of the conditional distribution holds) then, under mild technical assumptions,
the result from [9] can be applied to the progressive enlargement G.
Suppose that a pair (τ, P) is such that the hypothesis (H) is not satisfied by the F-conditional
distribution. It is then natural to ask whether there exists a probability measure P̄ equivalent P such
that the pair (P̄, τ ) satisfies the hypothesis (H), that is, whether there exists a probability measure
P̄ equivalent to P such that, for all u ≤ t,
F̄u,u := P̄ ( τ ≤ u | Fu ) = P̄ ( τ ≤ u | Ft ) =: F̄u,t .
(45)
Under the density hypothesis (see Section 4), the answer to this question is known to be positive
(see El Karoui et al. [15] and Grorud and Pontier [16]). By contrast, to the best of our knowledge,
this question remains open in the case of an honest time.
In this subsection, we address this question under the standing assumption that a pair (τ, P),
given on a filtered probability (Ω, G, F, P), is such that the F-conditional distribution is separable.
This means that the hypothesis (HP ) holds under P. Note, however, that the case of an honest time
is not covered by foregoing results since we assume here that Fu,t > 0. Let Gt = P ( τ > t | Ft ) and
Fu,t = P ( τ ≤ u | Ft ) = Yu Xt for u ≥ t.
Lemma 5.1. Assume that F-conditional distribution of τ under P is completely separable. Let the
process Z G be given by
ZtG = Zet 1{τ >t} + Zbτ,t 1{τ ≤t}
(46)
where Zbu,t =
Fu,u
Fu,t
and
Z
et = G−1
Z
1
−
t
Zbu,t dFu,t
(0,t]
Then the process Z
G
b
= G−1
1
−
E
Z
1
.
P
τ,t {τ ≤t} Ft
t
is a (P, G)-local martingale.
bu,u = 1 and thus it is trivially
Proof. The proof hinges on an application of Corollary 3.3. First, Z
an F-predictable process. Next, we need to check condition (i) in Corollary 3.3, which now reads:
the process (Wt )t≥0 is a (P, F)-local martingale where
Z
et Gt +
bu,u dFu = Z
et Gt + Ft .
Wt = Z
Z
(47)
(0,t]
We observe that
Z
et Gt + Ft = 1 + Ft −
Z
Zbu,t dFu,t
(0,t]
Z
= 1 + Xt Yt −
Z
Xu dYu = 1 + X0 Y0 +
(0,t]
Yu− dXu
(0,t]
so that the process W is indeed a (P, F)-local martingale.
Finally, we need to check condition (ii) in Corollary 3.3, which now becomes: for every u > 0,
the process (Wu,t = Xt Zbu,t )t≥u is a (P, F)-local martingale. To this end, we note that
Yu Xu
Fu,u
= Xt
= Xu ,
Wu,t = Xt Zbu,t = Xt
Fu,t
Yu Xt
(48)
which is obviously a (P, F)-local martingale for t ≥ u. In view of Proposition 3.1, we conclude that
Z G is a (P, G)-local martingale.
30
Progressive Enlargements and Semimartingale Decompositions
We will also need the following lemma.
Lemma 5.2. Assume that the hypothesis (HP ) holds under P. Then for any F-adapted, P-integrable
process X we have that, for every s ≤ t,
Fs,t EP (Xτ 1{τ ≤s} | Fs ) = Fs,s EP (Xτ 1{τ ≤s} | Ft ).
(49)
Proof. Note that equality (49) is trivially satisfied for X = 1. In general, it suffices to consider an
elementary F-adapted process of the form Xt = 1A 1[u,∞) (t) where u ∈ R+ and A ∈ Fu . Clearly
both sides of (49) vanish when u > s. For any u ≤ s, we obtain
Fs,t EP (1A 1[u,∞) (τ )1{τ ≤s} | Fs ) = 1A Fs,t EP (1{τ ≤s} − 1{τ <u} | Fs ) = 1A Fs,t (Fs,s − Fu−,s )
= 1A (Fs,t Fs,s − Fu−,t Fs,s ) = 1A Fs,s (Fs,t − Fu−,t ) = 1A Fs,s EP (1{τ ≤s} − 1{τ <u} | Ft )
= Fs,s EP (1A 1[u,∞) (τ )1{τ ≤s} | Ft )
where the hypothesis (HP ) was used in the third equality.
Proposition 5.1. Assume that:
(i) the conditional distribution of a random time τ under P is completely separable so that the
hypothesis (HP ) is satisfied by the pair (τ, P),
(ii) the process Z G given by formula (46) is a positive (P, G)-martingale with the property that
EP (ZtG | Ft ) = 1 for t ∈ R+ .
Then there exists an equivalent probability measure P̄ such that the hypothesis (H) holds under P̄.
Proof. It suffices to show that (45) holds under P̄, where the probability measure P̄ is defined on Gt
by dP̄|Gt = ZtG dP|Gt . We note that, for all u ≤ t,
bτ,u 1{τ ≤u} Fu = (Xu )−1 P Xτ 1{τ ≤u} Fu
F̄u,u = P ZuG 1{τ ≤u} Fu = P Z
= (Xt )−1 P Xτ 1{τ ≤u} Ft = P Zbτ,t 1{τ ≤u} Ft = P ZtG 1{τ ≤u} Ft = F̄u,t
where the fourth equality is an immediate consequence of Lemma 5.2. We conclude that the hypothesis (H) is satisfied by the pair (τ, P̄).
5.1.1
Hypothesis (H) in a General Set-up
In Sections 5.1.1 and 5.1.2, we summarize and rephrase the most pertinent results from Coculescu
et al. [9]. We are given a probability space (Ω, G, P) endowed with arbitrary filtrations F and G such
that F ⊂ G. Let H(P) stand for the class of all probability measures Q equivalent to P on (Ω, G)
such that F and G satisfy the hypothesis (H) under Q. Recall that the hypothesis (H) for F and G
under Q stipulates that any (Q, F)-local martingale is a (Q, G)-local martingale.
e ∈ H(P) and Q is a probability measure equivalent to Q
e on (Ω, G) such
Lemma 5.3. Assume that Q
dQ
that dQe is F∞ -measurable. Then Q belongs to H(P).
e ∈ H(P) there exists a probability
Lemma 5.4. Assume that H(P) is non-empty. Then for every Q
e
measure Q equivalent to Q on (Ω, G) and such that the following conditions are valid:
(i) the Radon-Nikodým density process ηt := ddQ
e |Gt is F-adapted,
Q
(ii) the probability measures Q and P coincide on the filtration F,
(iii) Q belongs to H(P),
(iv) every (P, F)-local martingale is a (Q, G)-local martingale.
Remark 5.1. Note that the probability measure P in Lemma 5.4 can be replaced by any probability
measure equivalent to P. This means that the assumption that H(P) is non-empty is quite strong:
it implies that for any probability measure P0 there exists a probability measure Q0 coinciding with
P0 on F and such that the hypothesis (H) holds under Q0 .
L. Li and M. Rutkowski
31
Remark 5.2. The property that the class H(P) is non-empty is not satisfied by F and the progressive
enlargement G when τ is an honest time, unless it is an F-stopping time (so that F = G and the
hypothesis (H) is trivially satisfied).
The next result provides another characterization of non-emptiness of the class H(P).
Proposition 5.2. The following conditions are equivalent:
(i) the class H(P) is non-empty,
(ii) there exists a probability measure Q equivalent to P on (Ω, G) such that every (P, F)-local martingale is a (Q, G)-local martingale.
5.1.2
Martingale Measures via Hypotheses (H) and (HP )
We assume that we are given an F-semimartingale X defined on a probability space (Ω, G, P) and
we denote by M(P, F) the class of all F-local martingale measures for X meaning that a probability
measure Q belongs to M(P, F) whenever: (i) Q equivalent to P on (Ω, G) and (ii) X is an (Q, F)-local
martingale.
Let G be a filtration such that F ⊂ G. We denote by M(P, G) the class of G-local martingale
measures for X. We assume that the class M(P, F) is non-empty and we search for sufficient
conditions under which the class M(P, G) is non-empty as well.
One possibility is to postulate that the hypothesis (H) holds under some probability measure
Q equivalent to P and to deduce that the hypothesis (H) is valid under some F-local martingale
measure. A particular example of an F-local martingale measure under which the hypothesis (H)
holds is produced using Lemma 5.4. Any F-local martingale measure under which the hypothesis
(H) holds is also a G-local martingale measure.
Proposition 5.3. (i) The classes M(P, F) and H(P) are non-empty if and only if the set M(P, F) ∩
H(P) is non-empty.
(ii) If the classes M(P, F) and H(P) are non-empty then the class M(P, G) is non-empty.
Remark 5.3. Note that the process X plays no essential role in the proof of Proposition 5.3 (except
for the assumption that the class M(P, F) is non-empty). Note also that the probability measure Q
constructed in the proof has the property that every (P0 , F)-local martingale is also an (Q, F)-local
martingale (since P0 and Q coincide on F) and thus a (Q, G)-local martingale (since Q ∈ H(P0 )).
Remark 5.4. The class M(P, F) can be replaced in part (i) of Proposition 5.3 by any subset P of
probability measures equivalent to P and such that: if P0 ∈ P and P00 = P0 on F then P00 ∈ P.
To summarize, the following conditions are equivalent:
(1) M(P, F) and H(P) are non-empty,
(2) M(P, F) ∩ H(P) is non-empty,
(3) M(P, F) is non-empty and there exists a probability measure Q equivalent to P on (Ω, G) such
that every (P, F)-local martingale is a (Q, G)-local martingale.
In view of Proposition 5.3 any of conditions (1)–(3) implies that the class M(P, G) is non-empty.
It is thus natural to refer to any of conditions (1)–(3) is the no-arbitrage condition.
Proposition 5.1 shows that if the pair (τ, P) is such that the complete separability of the conditional distribution holds then Proposition 5.3, which is due to Coculescu et al. [9], can be applied
to the progressive enlargement G.
5.2
Information Theory
For simplicity, we assume in this section that all (P, F)-martingales are continuous. The aim is
to apply the semimartingale decompositions developed in Section 4 to utility maximization and
information theory associated with continuous time models of financial markets (see, in particular,
32
Progressive Enlargements and Semimartingale Decompositions
Ankirchner and Imkeller [1]). Let us first recall the following definitions, borrowed from Ankirchner
and Imkeller [1].
Definition 5.1. A filtration F is said to be a finite utility filtration for a process V , if V is a
(P, F)-semimartingale with the semimartingale decomposition of the form
Z
Vt = Ut +
φu d hU, U iu
(50)
(0,t]
for some (P, F)-martingale U and a (P, F)-predictable process φ.
Remark 5.5. It was shown by Delbaen and Schachermayer in [10] that for a locally bounded
semimartingale V , the existence of an (local) martingale measure or equivalently if V satisfies the
No Free Lunch with Vanishing Risk (NFLVR) property then the decomposition of the process V
must be of the form given in (50).
Definition 5.2. Let V and F satisfy Definition 5.1. Suppose than K is a filtration such that F ⊂ K.
The K-predictable process ψ such that the process
Z
Ut −
ψu d hU, U iu
(0,t]
is a (P, K)-local martingale is called the information drift of K with respect to F.
Remark 5.6. Ankirchner and Imkeller also shown in [1] (see Proposition 1.2 therein) that the
difference of logarithmic utility of the process V between two filtrations F ⊂ K depends only on the
information drift of K with respect to F.
In the rest of this section, one work on the usual filtered probability space (Ω, G, P) with a filtration F satisfying the usual conditions and such that F is a finite utility filtration for a semimartingale
V . The progressive enlargement of F with a random time τ is denoted by G.
Under the assumption that all (P, F)-martingales involved are square-integrable (in order to apply
the Kunita-Watanabe decomposition), we aim to show that the progressive enlargement of F with a
random time τ satisfying the extended density hypothesis is once again a finite utility filtration for
V . We also compute the information drift of G with respect to F.
Lemma 5.5. Assume the random time τ satisfying the extended density hypothesis, then G is a
finite utility filtration for V .
Proof. By standing assumption, the filtration F is a finite utility filtration for the process V . Therefore, the process V has the following semimartingale decomposition
Z
Vt = Ut +
φu d hU, U iu
(51)
(0,t]
for some (P, F)-martingale U and (P, F)-predictable process φ. It is enough to show that the process
V is a (P, G)-semimartingale with the following decomposition
Z
Vt = Ūt +
φ̄u d Ū , Ū u
(52)
(0,t]
where Ū is some (P, G)-martingale and φ̄ a (P, G)-predictable process. Using Corollary 4.4 and the
assumption that all (P, F)-martingales are continuous, we deduce that the process Ū , defined by the
equality
Z
Z
−1
Ūt = Ut −
1{τ >u} (Gu ) d hU, M iu −
1{τ <u} (mτ,u )−1 d hU, mτ,u iu ,
(53)
(0,t]
(0,t]
33
L. Li and M. Rutkowski
is a (P, G)-martingale. Recall that Doob-Meyer decomposition of G is denoted as Gt = G0 +Mt −At .
We assume hereafter that the (P, F)-martingale M is square-integrable. Then, by an application of
the Kunita-Watanabe decomposition to M and U , we obtain
Z
bt
Mt =
ηu dUu + L
(54)
(0,t]
b is strongly
where η is some (P, F)-predictable process and a square-integrable (P, F)-martingale L
orthogonal to U . Using (54), one can write
d hU, M iu = ηu d hU, U iu .
In the next step, we assume for all u ≥ 0, the martingale (mu,t )t≥u is square integrable. Therefore,
once again by Kunita-Watanabe decomposition, we obtain
Z
mu,t =
ξu,s dUs + Lut
(55)
(u,t]
where (ξu,t )t≥0 is a family of (P, F)-predictable processes parametrized by u and Lu a family of
square integrable (P, F)-martingales strongly orthogonal to U for all u. Consequently, by combining
(54) and (55) with (53), we obtain
Z
Z
−1
Ūt = Ut −
1{τ >u} (Gu ) ηu d hU, U iu +
1{τ <u} (mτ,u )−1 ξτ,u d hU, U iu
(0,t]
(0,t]
Z
= Ut −
1{τ >u} (Gu )−1 ηu + 1{τ <u} (mτ,u )−1 ξτ,u d hU, U iu .
(0,t]
Continuity of (P, F)-martingale yields the equality hU, U i = Ū , Ū . Therefore, the canonical (P, G)semimartingale decomposition of U reads
Z
(56)
Ut = Ūt +
γτ,u d Ū , Ū u
(0,t]
where the (P, G)-predictable process γτ,u is given by
γτ,u := 1{τ >u} (Gu )−1 ηu + 1{τ <u} (mτ,u )−1 ξτ,u .
It is now easy to see that the (P, G)-semimartingale decomposition of V is of the form given by (52)
with φ̄t = φt + γτ,t .
Corollary 5.1. The information drift of the filtration G with respect to F is given by the following
expression
γτ,u = 1{τ >u} (Gu )−1 ηu + 1{τ <u} (mτ,u )−1 ξτ,u .
Proof. It is enough
to check
Definition 5.2. The asserted formula follows directly from representation
(56) and the fact Ū , Ū = hU, U i.
34
Progressive Enlargements and Semimartingale Decompositions
References
[1] Ankirchner, S. and Imkeller, P.: Financial markets with asymmetric information: information
drift, additional utility and entropy. In: Stochastic Processes and Applications to Mathematical
Finance: Proceedings of the 6th Ritsumeikan International Symposium, Akahori, J., Ogawa, S.
and Watanabe, S. (eds), World Scientific, 2007.
[2] Azéma, J.: Quelques applications de la théorie générale des processus I. Inventiones Mathematicae 18 (1972), 293?-336.
[3] Barlow, M.: Martingale representation with respect to expanded σ-fields. Working paper, 1977.
[4] Barlow, M.: Study of a filtration expanded to include an honest time. Zeitschrift für Wahrscheinlichkeitstheorie and Verwandte Gebiete 44 (1978), 307–323.
[5] Baudoin, F.: Modelling anticipations on financial markets. Paris-Princeton Lectures on Mathematical Finance 2002, Springer, 43–94. 2003
[6] Bielecki, T.R. and Rutkowski, M.: Credit Risk: Modeling, Valuation and Hedging. SpringerVerlag, Berlin Heidelberg New York, 2002.
[7] Brémaud, P. and Yor, M.: Changes of filtrations and of probability measures. Zeitschrift für
Wahrscheinlichkeitstheorie and Verwandte Gebiete 45 (1978), 269–295.
[8] Biagini, F. and Øksendal, B.: Minimal variance hedging for insider trading. International Journal of Theoretical and Applied Fiance Vol. 9, No. 8 (2006), 1351–1375.
[9] Coculescu, D., Jeanblanc, M. and Nikeghbali, A.: Default times, non arbitrage conditions and
change of probability measures. Working paper, 2009.
[10] Delbaen, F. and Schachermayer, W.: The existence of absolutely continuous local martingale
measures The Annals of Applied Probability. Vol. 5, No. 4 (1995) 926–945.
[11] Dellacherie, C.: Capacities and Stochastic Processes. Springer-Verlag, Berlin Heidelberg New
York, 1972.
[12] Dellacherie, C. and Meyer, P.-A.: A propos du travail de Yor sur le grossissement des tribus.
In: Séminaire de Probabilités XII, Lecture Notes in Mathematics 649. Springer-Verlag, Berlin
Heidelberg New York, 1978, pp. 70–77.
[13] Di Nunno, G. and Sjursen, S.: Information and optimal investment in defaultable assets. Working paper, 2010
[14] Elliott, R.J., Jeanblanc, M., and Yor, M.: On models of default risk. Mathematical Finance 10
(2000), 179–195.
[15] El Karoui, N., Jeanblanc, M., and Jiao, Y.: What happens after a default: the conditional
density approach. Working paper, 2009.
[16] Grorud, A. and Pontier, M.: Assymetrical information and incomplete markets. International
Journal of Theoretical and Applied Finance 4 (2001), 285-302.
[17] Guo, X., and Zeng, Y.: Intensity process and compensator: A new filtration expansion approach
and the Jeulin-Yor theorem. Annals of Applied Probability 18 (2008), 120–142.
[18] Jacod, J.: Grossissement initial, hypothèse (H 0 ) et théorème de Girsanov. In: Grossissements
de filtrations: exemples et applications, Lecture Notes in Mathematics 1118, T. Jeulin and M.
Yor (eds). Springer-Verlag, Berlin Heidelberg New York, 1985, pp. 15–35.
[19] Jacod, J. and Shiryaev, A.N.: Limit Theorems for Stochastic Processes. Springer-Verlag, Berlin
Heidelberg New York, 2003.
L. Li and M. Rutkowski
35
[20] Jeanblanc, M. and Le Cam, Y.: Progressive enlargement of filtration with initial times. Stochastic Processes and their Applications 119 (2009), 2523–2543.
[21] Jeanblanc, M. and Song, S.: An explicit model of default time with given survival probability.
Working paper, University of Evry, 2010.
[22] Jeanblanc, M. and Song, S.: Random times with given survival probability and their Fmartingale decomposition formula. Working paper, University of Evry, 2010.
[23] Jeanblanc, M., Yor, M., and Chesney, M.: Mathematical Methods for Financial Markets.
Springer-Verlag, Berlin Heidelberg New York, 2009.
[24] Jeulin, T.: Grossissement d’une filtration et applications. In: Séminaire de Probabilités XIII,
Lecture Notes in Mathematics 721. Springer-Verlag, Berlin Heidelberg New York, 1979, pp.
574–609.
[25] Jeulin, T.: Semi-martingales et grossissement d’une filtration. Lecture Notes in Mathematics
833. Springer-Verlag, Berlin Heidelberg New York, 1980.
[26] Jeulin, T.: Comportement de semi-martingales dans un grossissement de filtration. Zeitschrift
für Wahrscheinlichkeitstheorie and Verwandte Gebiete 52 (1980), 149–182.
[27] Jeulin, T. and Yor, M.: Grossissement d’une filtration et semi-martingales: formules explicites.
In: Séminaire de Probabilités XII, Lecture Notes in Mathematics 649. Springer-Verlag, Berlin
Heidelberg New York, 1978, pp. 78–97.
[28] Jeulin, T. and Yor, M.: Nouveaux résultats sur le grossissement des tribus. Annales Scientifiques
de l’É.N.S. 4e Série, tome 11/3 (1978), 429–443.
[29] Li, L., and Rutkowski, M.: Constructing random times through the multiplicative approach.
Working paper, University of Sydney, 2010.
[30] Meyer, P.-A.: Représentations multiplicatives de sousmartingales d’après Azéma. In: Séminaire
de Probabilités XIII, Lecture Notes in Mathematics 721. Springer-Verlag, Berlin Heidelberg New
York, 1979, pp. 240–249.
[31] Nikeghbali, A.: An essay on the general theory of stochastic processes. Probability Surveys 3
(2006), 345–412.
[32] Nikeghbali, A. and Yor, M.: Doob’s maximal identity, multiplicative decompositions and enlargements of filtrations. Illinois Journal of Mathematics 50 (2006), 791–814.
[33] Protter, P.: Stochastic Integration and Differential Equations, Second Edition, Version 2.1.
Springer-Verlag, Berlin Heidelberg New York, 2005.
[34] Revuz, D. and Yor, M.: Continuous martingales and Brownian motion 3rd edition SpringerVerlag, Berlin Heidelberg New York, 1999
[35] Yor, M.: Grossissement d’une filtration et semi-martingales: théoremes généraux. In: Séminaire
de Probabilités XII, Lecture Notes in Mathematics 649. Springer-Verlag, Berlin Heidelberg New
York, 1978, pp. 61–69.
[36] Yor, M.: Grossissement de filtrations et absolue continuité de noyaux. In: Grossissement de
filtrations: exemples et applications. Lecture Notes 1118. Th. Jeulin and M. Yor, eds. SpringerVerlag, Berlin Heidelberg New York, 1980, pp. 6–14.