Working Paper
Series
_______________________________________________________________________________________________________________________
National Centre of Competence in Research
Financial Valuation and Risk Management
Working Paper No. 588
The Mathematical Structure of Horizon-Dependence
in Optimal Portfolio Choice
Tahir Choulli
Martin Schweizer
First version: October 2009
Current version: October 2009
This research has been carried out within the NCCR FINRISK project on
“Mathematical Methods in Financial Risk Management”
___________________________________________________________________________________________________________
The mathematical structure of horizon-dependence
in optimal portfolio choice
Tahir Choulli
University of Alberta
Martin Schweizer ∗
ETH Zürich
Department of Mathematical
Departement Mathematik
and Statistical Sciences
632 Central Academic Building
ETH-Zentrum, HG G 51.2
Edmonton, AB T6G 2G1
CH – 8092 Zürich
Canada
[email protected]
Switzerland
[email protected]
Abstract:
For every T ≥ 0, we consider the classical problem of maximising expected
utility from terminal wealth for the time horizon T . This yields a collection
¡
¢
b = (X
bT )T ≥0 of optimal final wealths and a family ϑ̂ = ϑ̂(T, ·)
of optimal
X
T ≥0
b has
investment strategies, one each for every time horizon T . We prove that X
e which is a semimartingale, and that ϑ̂ has a version which is of
a version X
finite variation with respect to the horizon parameter T . With the help of a
Key words:
new stochastic Fubini theorem, we also establish a novel decomposition of the
e
semimartingale X.
optimal portfolio choice, horizon-dependence, utility maximisation, semimartingales, stochastic integrals, stochastic Fubini theorem, mathematical finance
MSC 2000 Classification Numbers:
60G48, 60H05, 60G07, 28B05, 91B28
This version: 26.10.2009
∗
corresponding author
0. Introduction
Optimal investment problems are ubiquitous. They come up as soon as people start thinking
about their future lifestyles in some structured way and try to achieve from their available
resources a subjectively optimal outcome. Literally hundreds of papers have studied the
¡
many facets of such questions, ranging from psychological aspects see Jaggia/Thosar (2000)
¢
for a randomly chosen recent sample over economic issues to purely mathematical problems.
But despite the vast amount of literature, one aspect of key importance seems not well
understood to date: How exactly does the planning horizon affect the optimal investment
decision? Everyone agrees that there is some effect, and awareness of this fundamental issue
can be traced back in the scientific literature at least to Hakansson (1969) or Yaari (1965),
to name just two early and well-known references. But a general quantitative result has
remained elusive.
In this paper, we provide a rigorous mathematical answer to the above question in a
very general setting. Our financial market is modelled by a general semimartingale, and we
consider for each T ≥ 0 the problem of maximising expected utility from terminal wealth for
b = (X
bT )T ≥0 of optimal final wealths and
the time horizon T . This leads to a collection X
¡
¢
a family ϑ̂ = ϑ̂(T, ·) T ≥0 of optimal investment strategies, one each for every time horizon
b and we show how the field ϑ̂ behaves as a
T . We elucidate the structure of the process X
function of T , thus exhibiting precisely the mathematical structure of horizon-dependence in
optimal portfolio choice.
To the best of our knowledge, the approach in this paper is entirely new, and the main
results have no genuine precursors in the literature. (For completeness, we should mention
that the case of logarithmic utility has been solved in full generality; see Goll/Kallsen (2003).
But this case is extremely special and non-typical — logarithmic utility is well known to
induce myopic behaviour, which means that there is no dependence on the time horizon at
all.) More detailed discussions on related work are provided later in Section 3 when we have a
precise mathematical problem formulation and the appropriate terminology at our disposal.
We are also well aware that many new questions (as well as potential applications) arise from
our work here, and several of these will be addressed in forthcoming papers.
The paper is structured as follows. Section 1 provides the mathematical formulation of
the problem, explains the three main results, and gives an overview of the ideas and methods
b has a version X
e which
needed for their proofs. Section 2 proves the first main result, that X
is a semimartingale. Section 3 proves the second main result, that ϑ̂ has a version which is of
finite variation with respect to the horizon parameter T . Section 5 establishes as third main
e This is based on a
result a remarkable and novel decomposition of the semimartingale X.
new stochastic Fubini theorem, which is of independent mathematical interest and presented
in Section 4.
1
1. Problem formulation and overview of results
In this section, we mathematically explain the basic problem studied in this paper. We
introduce some notation, provide intuition, and give an overview of the main results.
Let (Ω, F , P ) be a probability space with a filtration IF = (Ft )t≥0 on which we have
an IRd -valued semimartingale S = (St )t≥0 . We consider IRd -valued predictable processes
R
ϑ = (ϑt )t≥0 which are S-integrable in the sense that the stochastic integral ϑ . S = ϑ dS as a
process is well defined (and hence a real-valued semimartingale). For a fixed constant x > 0,
such a ϑ is called (an) admissible (strategy) up to time T ∈ [0, ∞] if for some a ≥ 0, we have
P -almost surely
(1.1)
Xtx,ϑ := x + ϑ . St = x +
Zt
0
ϑu dSu ≥ −a
for 0 ≤ t ≤ T .
The usual interpretation in mathematical finance is as follows. We consider a financial
market with (one riskless and) d risky assets evolving over time; so the random variable St
describes (discounted) asset prices at time t. It is well known that semimartingales are the
most general possible models for this if some very basic economic postulates on the financial
market should be satisfied. The assets can be traded dynamically over time; such activities
by an investor are modelled by some ϑ where ϑit is the number of shares of asset i that the
investor has decided to hold at time t. Trading must be done in a self-financing way so that
price fluctuations in S do not generate or require additional cashflows to keep ϑ going. This
translates mathematically into the fact that if the investor starts with initial capital x at
time 0, his wealth at time t is given by Xtx,ϑ from (1.1). (For a loose explanation, note that
ϑu dSu is the infinitesimal gain or loss at time u when the investor holds the portfolio ϑu
and prices move by dSu . By the self-financing condition, changes in wealth occur only in this
way; so the wealth at time t is obtained by adding, via the integral, to the initial wealth x
all infinitesimal changes.) Finally, the lower bound in (1.1) is a budget constraint designed
to eliminate economically unreasonable strategies (e.g., doubling). For a textbook account,
we refer to Delbaen/Schachermayer (2006) or Föllmer/Schied (2004).
Optimal portfolio choice is the problem of finding, for a given time horizon T < ∞, a
strategy with “best” final wealth XTx,ϑ . To quantify this, let U be a utility random field;
this is a mapping from [0, ∞) × IR × Ω to IR where we think of U (T, y, ω) as the investor’s
subjective quantitative assessment of having the amount y at time T in state ω. Naturally,
y 7→ U (T, y, ω) is thus assumed to be increasing and concave, and (ω, T ) 7→ U (T, y, ω) should
be a decent stochastic process. It is classical that the optimisation problem
(1.2) maximise the expected utility E[U (T, XTx,ϑ )] of final wealth over all admissible strategies
up to time T
2
has for each time horizon T a unique solution, under suitable (mild) assumptions on U (and
S); see for instance Kramkov/Schachermayer (1999) or Karatzas/Žitković (2003).
But how does all that depend on the time horizon? In mathematical terms, we have for
bT , the solution of (1.2), which is a stochastic integral of S. More
each T a random variable X
precisely, the integrand is the optimal strategy for the problem (1.2); so if we denote this by
¡
¢
ϑ̂(T, u) 0≤u≤T , we have
(1.3)
bT =
X
x,ϑ̂(T,·)
XT
=x+
ZT
ϑ̂(T, u) dSu .
0
bT )T ≥0 of optimal wealths — and this apparently
Our goal is to study the stochastic process (X
bT on T in (1.3). In
innocent question is quite difficult because of the double dependence of X
bT (ω) is not obvious at all and turns out to
fact, even getting some path regularity for T 7→ X
involve a lot of work.
For the purposes of this section’s overview, we are deliberately vague about the precise
assumptions on U (and S); full details are given in later sections when they are needed and
used. The only exception to such precision is that we shall simply assume that (1.2) has a
unique solution for each T , without further specifics and without concrete sufficient conditions
on U . That last point will be addressed in forthcoming work.
Our three main results clarify in the general semimartingale setting for S the mathematical structure of horizon-dependence in optimal portfolio choice. We prove:
b of optimal wealths has a version X
e which is a semimartingale. This
1) The process X
bT )T ≥0 can be aggregated into a nice
means that the collection of random variables (X
stochastic process.
¡
¢
2) The two-parameter process ϑ̂(T, u) T ≥0, 0≤u≤T of optimal strategies for (1.2) has a
version ϑe which is of finite variation with respect to T . This means that the optimal
strategy field depends nicely on the time horizon.
e of optimal wealths admits a unique decomposition as the sum of a
3) The semimartingale X
stochastic integral of S and a predictable process B̄ of finite variation: P -almost surely,
(1.4)
eT = x +
X
ZT
ϑ̄u dSu + B̄T
0
for each T ≥ 0,
where ϑ̄ = (ϑ̄u )u≥0 is an IRd -valued predictable S-integrable process which does not
depend on T .
The structure and above all existence of the decomposition (1.4) is quite remarkable.
eT is a stochastic integral with respect to S, with an
For each T , we know from (1.3) that X
3
e as a stochastic integral of one single
integrand depending on T . However, (1.4) gives us X
integrand ϑ̄, plus a correction term B̄. This kind of decomposition is conceptually reminiscent
of, but technically quite different from, the optional decomposition theorem due to Kramkov
e and the more detailed study of the horizon-dependence
(1996). For the further analysis of X
in (1.2), the representation (1.4) will play a key role.
The idea for proving 3) is at least conceptually quite simple. Since a good version ϑe of
T 7→ ϑ̂(T, ·) is by 2) of finite variation, we can write
(1.5)
eT = x +
X
ZT
0
e u) dSu = x +
ϑ(T,
ZT
0
e u) dSu +
ϑ(u,
ZT ZT
0
u
e
ϑ(dt,
u) dSu .
e ·). From the last term in (1.5), we expect that
Not surprisingly, ϑ̄ is then the diagonal of ϑ(·,
“ B̄T =
ZT Zt
0
0
e
dSu ϑ(dt,
u) ”
by formally applying Fubini. To prove that this is possible and that the last term in (1.5)
indeed defines a predictable process of finite variation, we establish in Section 4 a new stochastic Fubini theorem for (measure-valued) stochastic integrals, with respect to a martingale, of
measure-valued integrands. This generalises earlier work by several authors and constitutes
another new result which is of independent mathematical interest.
The proofs for 1) and 2) are considerably more complicated, and the more difficult result is 1). From classical theory, there are two ways of showing that a given process Y
is a semimartingale; see the textbook Protter (2005) for more background. One can try
to exhibit a decomposition of Y into a local martingale and a process of finite variation if
one knows the dynamics of Y sufficiently well. Alternatively, one can appeal to the deep
Bichteler–Dellacherie characterisation result for semimartingales if one can show that Y is a
good integrator. The latter means that as a functional on simple (i.e. piecewise constant)
predictable processes ψ, the stochastic integral ψ . Y (which is for such ψ defined in an elementary way) has a suitable continuity property. Our idea is to use a variant of the second
approach which has recently come up in a line of work going back to S. Ankirchner and
P. Imkeller: If one has an unbounded increasing function U (to be thought of as a utility
function) such that E[U (ψ . YT )] remains bounded over all simple predictable ψ, then Y is a
semimartingale (on [0, T ]); see Ankirchner (2005) and Ankirchner/Imkeller (2005). However,
there are two major obstacles. The first is that such a result is only available if the utility
U (T, y, ω) depends only on y and, more importantly, if Y is locally bounded. We have no such
b and we also do not want to restrict ourselves to unbounded utilities.
information on our X,
Thus we go back to Delbaen/Schachermayer (1994) who essentially already showed that a
locally bounded process is a semimartingale if the family of final values of all its stochastic
4
integrals over simple bounded admissible integrands is bounded in probability. We exploit
b as process of wealths to translate (in Proposition 2.10) simple admissible
the structure of X
b into (general, admissible) stochastic integrals over S. Then we
stochastic integrals over X
deal with the big jumps, and finally, an indirect argument establishes the semimartingale
b in Theorem 2.14.
property of X
But there is a second, fundamental obstacle. All the above results start from a process
Y which is adapted and RCLL (i.e. has trajectories which are right-continuous and admit left
bT )T ≥0 is just a family of random variables for which we have a priori
limits). In contrast, (X
no more information than the structure (1.3) for each T . So the very first thing we must do is
b has a version X
e with RCLL trajectories. This crucially exploits (1.3). We
to establish that X
bT is right-continuous in probability (Theorem 2.8), by adapting a similar
first show that T 7→ X
x,ϑ̂(T,·)
argument from Kramkov/Schachermayer (1999) who established continuity of XT
with
bT admits right and left limits
respect to x. We then prove (Proposition 2.11) that T 7→ X
along the rationals, combining to that end (in Lemma 2.1) classical ideas from martingale
theory and mathematical finance in a novel way. Putting all these results together finally
e of X,
b and then we can proceed
yields (Proposition 2.13) the existence of an RCLL version X
along the lines sketched above.
As is well known, stochastic integrals are often easier to handle when the integrator is
not a semimartingale, but a martingale. This is because one has for martingales and their
stochastic integrals an isometry property and several classical inequalities. To exploit that,
we show (in Proposition 2.7) how one can obtain the martingale property (and even very good
integrability properties) for S by passing to a well-chosen equivalent probability measure Qm
— at least locally, up to an arbitrarily large stopping time Tm . This important auxiliary
result is used in several places in the paper, by working under Qm instead of under P .
Finally, it remains to establish the result 2) concerning the structure of ϑ̂. To that end,
we view ϑ̂(T, ω, u) as an IRd -valued process indexed by T and defined on Ω̄ = Ω × (0, ∞), and
we aim at proving that it is a semimartingale. (Since the filtration on Ω̄ is chosen constant, the
b we have to start by obtaining some
finite variation property is then immediate.) Like for X,
bT , we first prove
regularity for ϑ̂. Similarly to the right-continuity in probability of T 7→ X
that T 7→ ϑ̂(T, ·) is right-continuous in measure (Lemma 3.2) and has a product-measurable
¡
¢
is a
version ϑe on [0, ∞) × Ω̄ (Lemma 3.3). This easily implies that the diagonal ϑ̂(T, T )
T ≥0
predictable S-integrable process (Corollary 3.6). To obtain the semimartingale property of ϑ̂
e ·) is (locally, under an equivalent
with respect to T , we prove (Proposition 3.10) that T 7→ ϑ(T,
measure Qm ) a quasimartingale on Ω̄, by first establishing (Proposition 3.7) that it has good
continuity properties as an integrator for piecewise constant integrands (with respect to T ,
over Ω̄). The quasimartingale property then gives the existence of a right-continuous version,
and the above continuity property in turn yields by the classical Bichteler–Dellacherie result
e ·) is a semimartingale.
that T 7→ ϑ(T,
5
2. The process of optimal wealths has a semimartingale version
In this section, we prove that the process
bt =
X
x,ϑ̂(t,·)
Xt
= x + ϑ̂(t, ·) . St = x +
Zt
0
ϑ̂(t, u) dSu ,
t≥0
e which is a semimartingale. This will
of optimal wealths over time from (1.3) has a version X
be done in Section 2.2, while Section 2.1 prepares the ground with some results of general
interest. The main idea and overall structure of the argument are explained in Section 1, but
the results are presented here in a different order for reasons of proof efficiency. Since T is
b
often used as a fixed time horizon in this section, we have changed the running index for X
from T to t.
2.1. Some general results
We always work on a filtered probability space (Ω, F , IF, P ) with IF = (Ft )t≥0 satisfying
W
the usual conditions of right-continuity and P -completeness, and set F∞ = t≥0 Ft . Given
a ≥ 0 and an arbitrary IRd -valued process Y = (Yt )t≥0 , a simple (a-)admissible integrand
I
P
hi I]]τi−1 ,τi ]] with I ∈ IN , each hi IRd -valued and
for Y is a process of the form H =
i=1
Fτi−1 -measurable and stopping times 0 ≤ τ0 ≤ τ1 ≤ · · · ≤ τI < ∞ such that the real-valued
I
R
P
τi
− Y τi−1 ) is uniformly bounded from
stochastic integral process H . Y = H dY =
htr
i (Y
i=1
below (by −a, respectively). If all the hi are constant vectors ci ∈ IRd , we call H very
simple. Like Delbaen/Schachermayer (1994), we say that Y satisfies (NFLVR) for (very)
simple admissible integrands if H n . YT → 0 in L0 for every T ∈ [0, ∞], whenever H n is for
each n a (very) simple εn -admissible integrand for Y and (εn )n∈IN decreases to 0. Moreover,
in analogy to Karatzas/Kardaras (2007), we say that Y satisfies (NUPBR) for (very) simple
a-admissible integrands if the family {H . YT | H (very) simple a-admissible integrand for Y }
of random variables is bounded in L0 for every T ∈ [0, ∞]. Note that the value of a > 0 is
not important for (NUPBR); all that matters is that we impose the same negative uniform
lower bound on H . Y for all H.
There is in Delbaen/Schachermayer (1994) also a notion of (NFLVR) with general integrands and we shall use that later. If S = (St )t≥0 is an IRd -valued semimartingale, we say
that S satisfies (NFLVR) if ϑn . ST → 0 in L0 for every T ∈ [0, ∞], whenever ϑn is for each
n an IRd -valued predictable S-integrable process which is εn -admissible for S (in the usual
sense that ϑn . S ≥ −εn ) and (εn )n∈IN decreases to 0. Note that we allow here general (as
opposed to simple) integrands, and so we have to assume a priori that S is a semimartingale.
6
Lemma 2.1. Suppose Y = (Yt )t≥0 is an adapted IRd -valued process and uniformly bounded
from below in each coordinate. If Y satisfies (NFLVR) for very simple admissible integrands
or (NUPBR) for very simple 1-admissible integrands, then Y admits with probability 1 right
and left limits along the rationals, i.e. there exists a set C of probability 1 such that for all
I + increases or decreases to some
ω ∈ C, lim Xrn (ω) exists in IR whenever (rn )n∈IN ⊆ Q
n→∞
r ∈ [0, ∞). If Y is progressively measurable, then Y even admits with probability 1 right and
left limits (i.e. along monotonic sequences (rn )n∈IN ⊆ IR+ ).
Proof. By working with each coordinate, we can and do assume without loss of generality
that Y is real-valued. We combine classical ideas from Dellacherie/Meyer (1982) and Delbaen/
Schachermayer (1994), and in particular follow in parts closely the proof of Theorem VI.48
in Dellacherie/Meyer (1982). Assume first only that Y is adapted. Define the processes
Ut := lim inf Yr ,
r&t, r∈I
Q
Vt := lim sup Yr ,
r&t, r∈I
Q
t≥0
and note that U and V are progressively measurable by Theorem IV.17 in Dellacherie/Meyer
(1978). To prove the P -a.s. existence of right limits along the rationals, we want to show that
the set {(ω, t) | Ut (ω) < Vt (ω)} is evanescent, or equivalently that for all rationals a < b, the
set E := {(ω, t) | Ut (ω) < a, Vt (ω) > b} is evanescent. This set is progressive, so its debut D
is a stopping time, and we want to reach a contradiction from assuming that P [D < ∞] 6= 0.
As in the proof of Theorem VI.48 in Dellacherie/Meyer (1982), let
A0 := {(ω, t) | D(ω) < t < D(ω) + 1, Yt (ω) < a}.
Then the measure of the projection of A0 on Ω is P [D < ∞] because D is the debut of E.
Choose an optional cross-section τ0 of A0 such that P [τ0 < ∞] > (1 − 12 )P [D < ∞]. Let
£
¢
¡
A1 := {(ω, t) | D(ω) < t < D(ω) + 12 ∧ τ0 (ω), Yt (ω) > b}
¤
note the small typo in Dellacherie/Meyer (1982)! , and choose an optional cross-section τ1
of A1 such that P [τ1 < ∞] > (1 − 14 )P [D < ∞]. Let
¡
¢
A2 := {(ω, t) | D(ω) < t < D(ω) + 14 ∧ τ1 (ω), Yt (ω) < a},
and choose an optional cross-section τ2 of A2 such that P [τ2 < ∞] > (1 − 18 )P [D < ∞],
etc. In this way, we construct stopping times τi with P [τi < ∞] > (1 − 2−(i+1) )P [D < ∞],
τi ≡ +∞ on {D = ∞}, τi < τi−1 on {τi < ∞}, and τi < D + 2−i on {τi < ∞}. Since then
P [τi = ∞, D < ∞] = P [D < ∞] − P [τi < ∞] ≤ P [D < ∞] 2−(i+1) ,
Borel-Cantelli implies that almost surely on {D < ∞}, we have τi < ∞ for all large i. So
there exists with probability 1 on {D < ∞} a random n0 ∈ IN such that (τi )i≥n0 decreases
to D and for i ≥ n0 , we have τ2i−1 < ∞ and τ2i < ∞, hence also Yτ2i ≤ a and Yτ2i−1 ≥ b.
7
Now we deviate from Dellacherie/Meyer (1982) and switch from martingale theory to
2n
P
cn I]]τ2i ,τ2i−1 ]]
mathematical finance. Take T = ∞, define very simple integrands H n :=
i=n
with constants cn ≥ 0 still to be chosen, and set Y∞ := 0. Then
Yτ2i−1 − Yτ2i
implies that
H
n.
Y∞ =
2n
X
i=n


0
= Y∞ − Yτ2i ≥ −a

Y
τ2i−1 − Yτ2i ≥ b − a
if τ2i = ∞
if τ2i < ∞, but τ2i−1 = ∞
if τ2i−1 < ∞
cn (Yτ2i−1 − Yτ2i )
2n
X
¯
ª
¯
I{τ2i−1 =∞,τ2i <∞} .
≥ cn (b − a)# i ∈ {n, . . . , 2n} τ2i−1 < ∞ + cn (−a)
©
i=n
The last sum is always 0 or 1 so that the second summand above is always at least −cn |a|.
For ω ∈ {D < ∞} and i ≥ n0 (ω), we have P -a.s. τ2i−1 (ω) < ∞; hence
H n . Y∞ (ω) ≥ cn (b − a)(n + 1) − cn |a|
for n ≥ n0 (ω), and this shows that if cn → 0 and (n + 1)cn → ∞, then
H n . Y∞ −→ +∞
(2.1)
P -a.s. on {D < ∞}.
The same argument applies for any t instead of ∞ if we truncate all the τi at t, i.e. replace
τi by τi ∧ t in the above computations.
For admissibility of H n , the same computations for Yτ2i−1 ∧t − Yτ2i ∧t lead to
H
n.
2n
X
¯
ª
¯
Yt ≥ cn (b − a)# i ∈ {n, . . . , 2n} τ2i−1 ≤ t + cn (Yt − a)
I{τ2i−1 >t,τ2i ≤t} .
©
i=n
The first term is ≥ 0 since cn ≥ 0, and the second term is 0 or cn (Yt − a). So if −y with y > 0
is the uniform lower bound for Y , we see from H n . Y ≥ −cn (y + |a|) =: −εn that each H n is
√
εn -admissible for Y , and εn & 0 if cn & 0. Choosing for instance cn = 1/ n + 1 therefore
produces with the corresponding (H n ) a sequence of very simple integrands that violates via
(2.1) both properties (NFLVR) and (NUPBR) if P [D < ∞] > 0; note that (εn ) is bounded
since it converges, and so we have a uniform lower bound for H n . Y simultaneously for all n.
This proves existence of right limits along the rationals, and left limits along the rationals
are obtained analogously, which ends the proof for the adapted case.
8
If Y is even progressive, the same arguments can be made for the processes
Ut0 := lim inf Ys ,
s&t
Vt0 := lim sup Ys ,
s&t
t ≥ 0,
since these are then progressive by Theorem IV.33 of Dellacherie/Meyer (1978). This yields
the second assertion.
q.e.d.
The next result is very simple but useful.
Lemma 2.2. If a process Y admits right limits along the rationals and is right-continuous
in probability, then Y has a version Ye with right-continuous trajectories.
Proof. For each t ≥ 0, set Yet :=
lim
r&t, r∈I
Q
Yr ; this is well defined P -a.s. by the assumption
that Y admits right limits along the rationals. For any t ≥ 0 and rationals rn & t, we get
Yrn → Yt in L0 by right-continuity in probability, and Yrn → Yet P -a.s. by definition; so
Yet = Yt P -a.s. and Ye is a version of Y . By construction, Ye is P -a.s. right-continuous. q.e.d.
The next result is very useful for establishing the semimartingale property of a process,
in general as well as later in Theorem 2.14.
Lemma 2.3. Suppose Y = (Yt )t≥0 is a real-valued adapted locally bounded process and fix
T ∈ (0, ∞). If Y is not a semimartingale, then one can find a sequence of simple predictable
processes |H n | ≤ 1 such that all the H n are c-admissible for Y for some c > 0 and the
sequence (H n . YT )n∈IN is not bounded in L0 .
Proof. Our argument follows very closely the work done by F. Delbaen and W. Schachermayer in Section 7 of Delbaen/Schachermayer (1994), but the result itself is not explicitly
there. By stopping, we can assume without loss of generality that Y is bounded, and we can
also normalise so that |Y | ≤ 1. It is clear that if we then construct the H n to be uniformly
bounded (in n as well), we can normalise that bound to 1, too.
Now we start with the set
G :=
(m−1
X
k=0
(Yτk+1
¯
)
¯
¯
− Yτk )2 ¯ m ∈ IN, 0 ≤ τ0 ≤ τ1 ≤ · · · ≤ τm ≤ T stopping times
¯
and, for a given element of G, put H := −2
and H . Yt =
m−1
P
k=0
m−1
P
k=0
Yτk I]]τk ,τk+1 ]] . Then |H| ≤ 2 since |Y | ≤ 1,
(Yτk+1 ∧t − Yτk ∧t )2 − Yτ2m ∧t + Yτ20 ∧t ≥ −1 shows that H is 1-admissible for
9
Y . If G is not bounded in L0 , we can thus take a sequence (H n ) of the above form such that
(H n . YT )n∈IN ⊆ G is not bounded in L0 , and we are already done.
So suppose from now on that G is bounded in L0 . To follow the rest of the proof, a copy
of Section 7 of Delbaen/Schachermayer (1994) should be ready to hand. Start there with
the proof of Theorem 7.2, assuming that Y (called X there) is not a semimartingale, and
produce a first sequence (H n ) as on page 506, lines 3–10. Use then that G is bounded in L0
to produce a new sequence (K n ) as on page 506, line 20. Go on to construct a next sequence
e n . YT )n∈IN is then unbounded in L0 . Note also
e n ) as on page 507, line 3, and note that (K
(K
e n | ≤ 1 for all n.
that |K
Now go to the proof of Lemma 7.3 in Delbaen/Schachermayer (1994) on page 505, and
recall that H there is just some family of simple predictable processes bounded by 1, like
e n instead of the H n in Delbaen/Schachermayer
e n , n ∈ IN , from above. Use these K
the K
¡
e n I]]0,T ]] . These
(1994) on page 505, line 2; construct Tn0 , Un and then Tn , and set H̄ n := K
n
H̄
n
n
correspond to the H I]]0,Tn ]] on page 505, lines 11 and 12, of Delbaen/Schachermayer
¢
(1994). Then property (a) on page 505, line 11, dropping δn , shows that (H̄ n . YT )− , n ∈ IN ,
is bounded; but even more, the estimate on page 505, line 7, and Tn ≤ Un imply that
H̄ n . Y ≥ −K − 2 so that each H̄ n is (K + 2)-admissible for Y . Finally, property (b) on page
505, line 12, with cn → ∞ and dropping δn shows that (H̄ n . YT )n∈IN is not bounded in L0 .
So up to normalising, these H̄ n give our sequence.
q.e.d.
At this point, we have to introduce our assumptions on the basic process S. As explained in part I of the textbook Delbaen/Schachermayer (2006), the standard approach in
mathematical finance would be to suppose that S is a semimartingale and satisfies (NFLVR).
The latter condition is motivated from economic postulates, and by the fundamental theorem
of asset pricing, it is equivalent to the existence of an equivalent σ-martingale measure for
S, which is a probability Q equivalent to P such that S becomes a Q-σ-martingale (i.e. a
stochastic integral S = ψ . M , with a one-dimensional integrand ψ > 0, of an IRd -valued
local Q-martingale M ); see Delbaen/Schachermayer (1998). Also equivalently, there exists
¯
¯
a strictly positive P -martingale Z up to ∞ (namely, the process of densities Zt = dQ
dP Ft ,
t ∈ [0, ∞], of Q with respect to P along IF ) such that the product ZS is a P -σ-martingale
on [0, ∞); this is just rewriting things from Q to P via the Bayes rule.
Our conditions on S will be slightly different. We also assume that S is a semimartingale,
and we introduce for fixed x > 0 and each stopping time τ the set
¯
©
Xτx := Xτx,ϑ = x + ϑ . Sτ ¯ ϑ is IRd -valued, predictable, S-integrable
ª
and x-admissible for S up to time τ .
Then (NFLVR) is equivalent to the combination of the following two conditions:
0
intersects L0+ only in 0.
(2.2) X∞
10
(2.3) For each (deterministic) T ∈ [0, ∞) and some (or equivalently all) x > 0, the set XTx is
bounded in L0 .
The above equivalence can be found in Section 3 of Delbaen/Schachermayer (1994) or more
succinctly in Lemma 2.2 of Kabanov (1997). The property (2.2) is usually called (NA) for no
arbitrage, while (2.3) has been blandly called BK in Kabanov (1997) or labelled (NUPBR),
for no unbounded profit with bounded risk , in Karatzas/Kardaras (2007).
Instead of (2.3), we consider
(2.4) For each bounded stopping time τ and some (or equivalently all) x > 0, the set Xτx is
bounded in L0 .
We also introduce the localised version
(2.5) There exists a sequence (σk )k∈IN of stopping times increasing to ∞ and such that for
each bounded stopping time τ and some (or equivalently all) x > 0, the set Xτx∧σk is
bounded in L0 for each k ∈ IN .
Instead of (NFLVR), we shall need a variant of the existence of an equivalent σ-martingale
measure Q for S as above. More precisely, a σ-martingale density for S (and P ) is a local
(P -)martingale Z = (Zt )t≥0 with Z0 = 1 and Z ≥ 0 and such that ZS is a (P -)σ-martingale.
We denote by Da,σ (S, P ) the family of all σ-martingale densities Z for S and P , and by
Za,σ (S, P ) the subset of all those Z with the additional property that Z log Z is locally
(P -)integrable; so the semimartingale Z log Z is then P -special which means that its running
supremum process (Z log Z)∗t = sup |Zs log Zs |, t ≥ 0, is locally P -integrable. Finally we
0≤s≤t
introduce De,σ (S, P ) := {Z ∈ Da,σ (S, P ) | Z > 0} and
Ze,σ (S, P ) := Za,σ (S, P ) ∩ De,σ (S, P ) = {Z ∈ Za,σ (S, P ) | Z > 0}.
Our main condition on S is that Ze,σ (S, P ) is nonempty, and this is simultaneously a bit
stronger and a bit weaker than the existence of an equivalent σ-martingale measure Q for
S. Indeed, the density process Z Q would have stronger global properties, because it is a
true P -martingale up to ∞ while Z ∈ Ze,σ (S, P ) is only a local P -martingale; but the local
integrability of Z log Z for Z is more than what we have for Z Q .
Let us briefly clarify how the above concepts and conditions are related. To that end,
we first recall two facts that will be used several times in the sequel:
(2.6) If Y is an IRm -valued σ-martingale and ψ is an IRm -valued predictable and Y -integrable
R
process, then also ψ . Y = ψ dY is a (real-valued) σ-martingale.
(2.7) Any real-valued σ-martingale uniformly bounded from below is a local martingale. If
its initial value is in L1 (e.g. if it is null at 0), then it is also a supermartingale.
The result in (2.6) can be found in Lemma 3.3 of Kallsen (2003) or Proposition III.6.42 of
Jacod/Shiryaev (2003). The first assertion of (2.7) follows directly from Corollaire 3.5 of
Ansel/Stricker (1994); the second one is then a well-known consequence of Fatou’s lemma.
11
Lemma 2.4. For an IRd -valued semimartingale S, conditions (2.4) and (2.5) are equivalent,
and follow from the condition that De,σ (S, P ) is nonempty.
Proof. To get (2.5) from (2.4), it is enough to take σk = k. For the converse, note that
(2.8)
P [Xτx,ϑ ≥ α] = P [Xτx,ϑ ≥ α, τ ≤ σk ] + P [Xτx,ϑ ≥ α, τ > σk ]
≤ P [Xτx,ϑ
∧σk ≥ α] + P [σk < τ ].
If ϑ is x-admissible for S up to τ , it is also x-admissible for S up to τ ∧ σk . So after we take
the supremum over all the former ϑ, the first term on the right-hand side of (2.8) goes to 0
for each fixed k as α → ∞, due to (2.5). Since σk % ∞ and τ is bounded, the second term
P [σk < τ ] can be made arbitrarily small for k large. Hence (2.4) follows from (2.5).
Finally we show that De,σ (S, P ) 6= ∅ implies (2.5). Take Z ∈ De,σ (S, P ) and a localising sequence (σk )k∈IN such that each Z σk is a uniformly integrable P -martingale. Then
setting dQk := Zσk dP defines a probability measure Qk equivalent to P since Z > 0, and
Z σk S σk = (ZS)σk is like ZS a P -σ-martingale. By the Bayes rule, this means that S σk is
a Qk -σ-martingale; see Proposition 5.1 of Kallsen (2003). For any ϑ which is S-integrable,
hence S σk -integrable, ϑ . S σk is thus a Qk -σ-martingale by (2.6); so if ϑ is also x-admissible
for S up to σk , then ϑ . S σk is a Qk -supermartingale by (2.7). For such ϑ, therefore,
EQk [x + ϑ . Sτ ∧σk ] = EQk [x + ϑ . Sτσk ] ≤ x
for any bounded stopping time τ .
This shows that Xτx∧σk is bounded in L1 (Qk ), hence also in L0 (Qk ), which equals L0 (P )
because Qk is equivalent to P .
q.e.d.
Our key assumption on S is now that S is an IRd -valued semimartingale and
the set Ze,σ (S, P ) is nonempty.
(2.9)
This clearly implies De,σ (S, P ) 6= ∅ and hence (2.4) by Lemma 2.4.
Our next auxiliary result is similar to Lemma 3.2 of Delbaen/Schachermayer (1994).
Recall that X x,ϑ = x + ϑ . S ≥ 0 if ϑ is x-admissible for S.
Lemma 2.5. Suppose that the IRd -valued semimartingale S satisfies (2.4). For any fixed
T ∈ (0, ∞), the family
¯
½
¾
¯
x,ϑ ¯
WT := sup Xt ¯ ϑ predictable and x-admissible for S, τ ≤ T stopping time
0≤t≤τ
is then bounded in L0 . As a consequence, W (T ) := ess sup WT is finite P -a.s.
n
Proof. Suppose WT is not bounded in L0 . Take a sequence W n = sup Xtx,ϑ in WT and
0≤t≤τn
n
some δ > 0 such that P [W > n] ≥ δ for all n. With the stopping time
¯
©
ª
n
σn := inf t ≥ 0 ¯ Xtx,ϑ > n ∧ τn ≤ T,
12
we clearly have
©
ª ©
ª ©
ª
n
n
n
n
> n ⊆ Xσx,ϑ
≥ n or Xτx,ϑ
> n ⊆ Xσx,ϑ
{W n > n} ⊆ {σn < τn } ∪ Xτx,ϑ
≥n .
n ∧τn
n
n
n
So if we set ϕn := ϑn I]]0,σn ∧τn ]] , we get x + ϕn . S ≥ 0 since ϑn is x-admissible, and
P [x + ϕn . ST ≥ n] ≥ P [W n > n] ≥ δ.
But this means that the family {x + ϕ . ST | ϕ is predictable and x-admissible} ⊆ XTx is not
bounded in L0 , which contradicts the property (2.4) for S.
q.e.d.
The next result is an already announced key technical tool in our analysis. It allows us to
obtain good martingale and integrability properties by a suitable change of measure combined
with stopping. For its formulation, we first need some notations, and we start by recalling
the set Za,σ (S, P ) from above. If the local P -martingale Z ≥ 0 with Z0 = 1 is written as
Z = E(N ) with a local (P -)martingale N null at 0 and such that 1 + ∆N ≥ 0, we define
¢
P ¡
(1 + ∆Ns ) log(1 + ∆Ns ) − ∆Ns . For Z ∈ Za,σ (S, P ),
the process V (N ) := 12 hN c i +
0<s≤ ·
the process V (N ) is locally (P -)integrable; the entropy-Hellinger process of Z with respect
to P is then the dual predictable projection of V (N ) with respect to P , and we denote this
predictable RCLL increasing process by hE (Z, P ). Finally, the minimal entropy-Hellinger
σ-martingale density for (S, P ) is that element Z ∗ of Za,σ (S, P ) – if it exists – which minimises
Z→
7 hE (Z, P ) over Z ∈ Za,σ (S, P ) in the sense of the strong order, i.e. hE (Z ∗ , P ) − hE (Z, P )
is increasing for all Z ∈ Za,σ (S, P ). If Z ∗ is a P -uniformly integrable P -martingale, the
∗
dP is called the minimal entropycorresponding probability measure Q∗ with dQ∗ = Z∞
Hellinger σ-martingale measure for (S, P ). More details can be found in Choulli/Stricker
(2005, 2006).
In several of our proofs, we have to change from P to an equivalent probability measure
Q which endows S with better properties. While L0 , hence also (2.4), and the semimartingale property of S are invariant under any such change of measure, condition (2.9) that
Ze,σ (S, P ) 6= ∅ involves a local integrability condition and is thus more delicate. It is there-
fore quite remarkable that we have the following stability result.
Theorem 2.6. Let S be an IRd -valued semimartingale and suppose that the probability
measure Q is equivalent to P . If Ze,σ (S, P ) 6= ∅ and if the density dQ
dP is bounded, then also
Ze,σ (S, Q) 6= ∅.
The proof of Theorem 2.6 is too long to be given here; it takes a paper on its own,
and so we refer to Choulli/Schweizer (2010) for details. However, we point out that all the
difficulties come from the fact that the stochastic logarithm N of a strictly positive local
P -martingale Z = E(N ) can have jumps whose size comes arbitrarily close to −1, so that
13
the term log(1 + ∆N ), which appears in the definition of V (N ), is hard to control. If the
filtration IF does not contain any discontinuous martingales (e.g., if it is generated by some
possibly multidimensional Brownian motion), this problem does not occur, and the result in
Theorem 2.6 is then obvious.
Here is then the announced technical result for good measure changes.
Proposition 2.7. Suppose that the IRd -valued semimartingale S satisfies (2.9). Let W be
any nonnegative random variable with W < ∞ P -a.s. For any fixed T ∈ (0, ∞), the following
hold:
1) There exists a probability measure Q ≈ P with bounded density
(2.10)
µ
2
2
exp W + sup |∆St | + sup |St |
0<t≤T
0≤t≤T
2
¶
dQ
dP
such that
∈ L1 (Q)
and the minimal entropy-Hellinger σ-martingale density Ze for (S T , Q) exists.
2) There exist a sequence of stopping times (Tm )m∈IN increasing stationarily to T and
a sequence of probability measures Qm ≈ P with Qm = Qm+1 on FTm such that for each
m ∈ IN , S Tm is a Qm -square-integrable Qm -martingale and W ∈ L2 (Qm ). (As a matter of
fact, dQm = ZeTm dQ, as can be seen in the proof.)
Proof. Since the arguments for Proposition 2.7 are not used in the sequel, this proof is
relegated to the Appendix.
q.e.d.
b
2.2. The semimartingale property of X
We now fix x > 0 and study in more detail the process
bt =
X
x,ϑ̂(t,·)
Xt
=x+
Zt
ϑ̂(t, u) dSu ,
0
t≥0
of optimal wealths in order to prove that it admits a version which is a semimartingale. A
very rough outline of the argument is as follows:
b is right-continuous in probability.
1) We first prove that X
b satisfies (NUPBR) for simple integrands which are x-admissible
2) We then show that X
b and deduce from Lemmas 2.1 and 2.2 that X
b has an RCLL version X.
e
for X,
e the big jumps to get a locally bounded process X̄. Assuming that
3) We subtract from X
X̄ is not a semimartingale, we use Lemma 2.3 to construct a sequence of strategies that
will be shown to contradict the property (2.4) of S.
Steps 2) and 3) also require an auxiliary result, where in particular Proposition 2.7 turns out
to be very useful.
14
Before embarking on that route, we make precise our assumptions on the utility field
U . Recalling that (t, y, ω) 7→ U (t, y, ω) is a mapping from [0, ∞) × IR+ × Ω to IR, we
assume that for fixed (ω, t), y 7→ U (t, y, ω) is a standard utility function on IR+ , i.e. strictly
concave, strictly increasing, in C 1 , and satisfying the Inada conditions Uy (t, 0, ω) = +∞,
Uy (t, ∞, ω) = 0. In addition, we shall need:
(2.11) The processes (ω, t) 7→ U (t, 0, ω)I{U (t,0,ω)>−∞} and (ω, t) 7→ U (t, b, ω), for any constant b > 0, are adapted, right-continuous and of locally integrable variation.
This condition is easy to verify if there is for example no dependence on ω. We also need:
(2.12) For any uniformly bounded sequence of stopping times (τn )n∈IN decreasing to some
stopping time τ and any a, b with 0 < a ≤ b < ∞, we have
lim sup |U (τn , y, ω) − U (τ, y, ω)| = 0
n→∞ a≤y≤b
P -a.s.
This essentially says that U is right-continuous with respect to t, locally uniformly in y.
Finally, as a joint condition on U and S, we need the earlier mentioned basic assumption
that the utility maximisation problem is well posed in the following sense:
(2.13a) For each bounded stopping time τ , the problem of maximising the expected utility
E[U (τ, g)] over all g ∈ Xτx has a finite value and a unique solution ĝ.
We then take an optimal strategy denoted by ϑ̂(τ, ·) so that
¢¤
£ ¡
sup E[U (τ, g)] = E[U (τ, ĝ)] = E U τ, Xτx,ϑ̂(τ,·) < ∞,
(2.13b)
g∈Xτx
x,ϑ̂(τ,·)
bτ := ĝ = Xτ
. The combination of (2.13a) and (2.13b) is denoted by (2.13),
and we set X
and this assumption deserves a few comments. First of all, it is reasonable in view of our
problem — if we can achieve infinite expected utility for some time horizon, something is
wrong with our model at a fundamental level. Secondly, there are known sufficient condi-
tions on U and S in a number of cases; see for instance Kramkov/Schachermayer (1999) or
Karatzas/Žitković (2003). Finally, note that while the optimal wealth ĝ is naturally unique
by the strict concavity of U with respect to y, there can be several strategies leading to ĝ.
We take here for ϑ̂(τ, ·) one of them, and we shall come back to the issues of nonuniqueness
and of a good choice later in Section 3.
b is right-continuous in probability. One inspiration why such
We now first prove that X
a result could hold comes from Kramkov/Sı̂rbu (2006) who showed that with respect to x,
x,ϑ̂(t,·)
Xt
x,ϑ̂(t,·)
Xt
is even differentiable (if U is independent of (ω, t) and C 2 with respect to y). Since
Rt
= x + ϑ̂(t, u) dSu depends on t in a much more complicated way than on x, our
0
result is conceptually quite different from that of Kramkov/Sı̂rbu (2006), but we can still
15
explain the intuition behind it. Increasing t (or τ ) a little has the effect of adding a stochastic
integral, which is a right-continuous operation since the stochastic integral is, and this rightcontinuity is then preserved thanks to the assumptions (2.11) and (2.12) on U . Formally, our
proof is modified from Lemma 3.6 in Kramkov/Schachermayer (1999), which establishes a
similar continuity result (but also with respect to x) for the solution of the dual problem.
Theorem 2.8. Suppose that the IRd -valued semimartingale S satisfies (2.9), and U satisfies
(2.11)–(2.13). Let (τn )n∈IN be any sequence of stopping times bounded by some T ∈ (0, ∞).
bτ in L0 .
bτ )n∈IN converges to X
If (τn ) decreases to a stopping time τ , then (X
n
Proof. 1) Take Z ∈ De,σ (S, P ) which is nonempty due to (2.9). As in the proof of Lemma
2.4, let (σk )k∈IN be a localising sequence for Z, define Qk by dQk := Zσk dP and recall that
S σk is then a Qk -σ-martingale. For any ε > 0,
¤
¤
£
¤
£
£
bτ − X
bτ − X
b τ | ≥ ε = P |X
bτ | ≥ ε, τn ≤ σk + P |X
bτ | ≥ ε, τn > σk
bτ − X
P |X
n
n
n
£
¤
bτ ∧σ − X
bτ ∧σ | ≥ ε + P [σk < τ1 ]
≤ P |X
n
k
k
because (τn ) is decreasing, and since σk % ∞ and τ1 is bounded, the last summand can be
bτ in L0 follows if we
bτ → X
made arbitrarily small by making k large. So the convergence X
n
bτ ∧σ in L0 as n → ∞. But in view of the definition
bτ ∧σ → X
show for each fixed k that X
n
k
k
b and because
= x + ϑ . S%∧σk = x + ϑ . S%σk for each stopping time %, this means
of X
that we can in effect argue for each fixed k as if we were working with S σk instead of S.
2) To simplify notations in the rest of the proof, we now fix k and write %0 := % ∧ σk for
each stopping time %. As seen in step 1), the process S 0 := S σk is a Qk -σ-martingale, or Qk
x,ϑ
X%∧σ
k
is an equivalent σ-martingale measure for S 0 . By the fundamental theorem of asset pricing,
this means that S 0 satisfies (NFLVR); see Delbaen/Schachermayer (1998).
3) For each bounded stopping time %, define the set
C(%0 , x) := X%x0 − L0+ (F%0 )
:= {x + ϑ . S%0 − Y | ϑ is x-admissible for S on [[0, %0 ]], Y ∈ L0+ (F%0 )}
= {x + ϑ . S%0 0 − Y | ϑ is x-admissible for S 0 on [[0, %0 ]], Y ∈ L0+ (F%0 )}.
As in Proposition 3.1 of Kramkov/Schachermayer (1999), each C(%0 , x) is convex and closed
T
C(%0n , x) = C(%0 , x) if (%n ) decreases to %;
in L0 because S 0 satisfies (NFLVR). Moreover,
n∈IN
T
indeed, each g from the intersection is measurable with respect to
F%0n = F%0 , and for
n∈IN
any equivalent σ-martingale measure Q for S and any ϑ admissible for S 0 , the process ϑ . S 0
is by (2.7) a Q-supermartingale so that
0
g = EQ [g | F%0 ] = EQ [x + ϑn . S%0 0n − Y n | F%0 ] ≤ x + ϑn . S%0 0 − EQ [Y n | F%0 ].
16
0
x,ϑ̂(% ,·)
b%0 because U is
Finally, the maximiser of E[U (%0 , g)] over g ∈ C(%0 , x) must be X%0
=X
strictly increasing with respect to y.
bτ 0 ) does not converge to X
bτ 0 in L0 so that for some δ > 0,
4) Suppose now that (X
n
¤
£
bτ 0 | ≥ δ ≥ δ for infinitely many n. As in the proof of Lemma 3.6 of
bτ 0 − X
we have P |X
n
bτ 0 ) so that ḡn ∈ C(τn0 , x), and use
bτ 0 + X
Kramkov/Schachermayer (1999), define ḡn := 12 (X
n
bτ 0 ) + 1 U (τn0 , X
bτ 0 ) for all
strict concavity of U with respect to y to get U (τn0 , ḡn ) ≥ 1 U (τn0 , X
2
n, as well as
n
2
¤
£
bτ 0 ) + 1 U (τn0 , X
bτ 0 ) + η ≥ η
P U (τn0 , ḡn ) ≥ 12 U (τn0 , X
2
n
for some η > 0 and for infinitely many n. By passing to a subsequence (gn ) that we again
index by n for ease of notation, we thus obtain
bτ 0 )] + 1 E[U (τn0 , X
bτ 0 )] + η 2 ≥ E[U (τn0 , X
bτ 0 )] + η 2
E[U (τn0 , gn )] ≥ 12 E[U (τn0 , X
2
n
bτ 0 is optimal in C(τ 0 , x) ⊇ C(τ 0 , x). But as we shall argue below in step
for all n, because X
n
n
bτ 0 )] = E[U (τ 0 , X
bτ 0 )], and so
5), assumption (2.12) implies that lim E[U (τn0 , X
n→∞
bτ 0 )] + η 2 .
lim inf E[U (τn0 , gn )] ≥ E[U (τ 0 , X
(2.14)
n→∞
Again by step 5) below, lim E[U (τn0 , gn ) − U (τ 0 , gn )] = 0, and because we always have
n→∞
lim inf (an + bn ) ≥ lim inf an + lim inf bn , we obtain from (2.14) that
n→∞
n→∞
n→∞
bτ 0 )] + η 2 .
lim inf E[U (τ 0 , gn )] ≥ E[U (τ 0 , X
(2.15)
n→∞
Now since all the gn are nonnegative, the standard Komlós trick from Lemma A1.1
of Delbaen/Schachermayer (1994) gives for all n ∈ IN some g̃n ∈ conv(gn , gn+1 , . . .) with
(g̃n )n∈IN converging P -a.s. to some g̃∞ . By the closedness in L0 of each C(%0 , x), this g̃∞ lies
T
C(τn0 , x) = C(τ 0 , x) since like the gn , gn+1 , . . ., each g̃n lies in the convex set C(τn0 , x).
in
n∈IN
The concavity of U with respect to y yields E[U (τ 0 , g̃n )] ≥ inf E[U (τ 0 , gm )], and since the
m≥n
last term is monotonic in n, letting n → ∞ and using (2.15) yields
bτ 0 )] + η 2 .
lim inf E[U (τ 0 , g̃n )] ≥ lim inf E[U (τ 0 , gn )] ≥ E[U (τ 0 , X
n→∞
n→∞
Again using (2.12), we get from step 5) below that lim E[U (τ 0 , g̃n )] = E[U (τ 0 , g̃∞ )]. Hence
n→∞
0
we find that g̃∞ ∈ C(τ , x) satisfies
bτ 0 )] + η 2 ,
E[U (τ 0 , g̃∞ )] = lim E[U (τ 0 , g̃n )] ≥ lim inf E[U (τ 0 , gn )] ≥ E[U (τ 0 , X
n→∞
n→∞
bτ 0 )n∈IN converges to X
bτ 0 in L0 .
bτ 0 . Thus (X
which clearly contradicts the optimality of X
n
17
5) It remains to argue the convergence assertions left open in step 4). For brevity,
0
. Suppose (Yn )n∈IN is any sequence of nonnegative random variables with
write τ 0 =: τ∞
U (τn0 , Yn ) ∈ L1 (P ) and U (τ 0 , Yn ) ∈ L1 (P ) for each n ∈ IN . Then we claim that
£
¤
(2.16)
lim E |U (τn0 , Yn ) − U (τ 0 , Yn )| = 0.
n→∞
Similarly, if U (τ 0 , Yn ) ∈ L1 (P ) for each n ∈ IN ∪ {∞} and Yn → Y∞ P -a.s., we claim that
£
¤
(2.17)
lim E |U (τ 0 , Yn ) − U (τ 0 , Y∞ )| = 0.
n→∞
bτ 0 and Yn := gn for (2.16), and Yn := g̃n for (2.17); they
Examples used in step 4) are Yn := X
can all be assumed without loss of generality to satisfy the required integrability. Indeed, we
¢−
¡
∈ L1 (P ) because we are maximising expected utility, and then
may suppose U (τn0 , Yn )
¢+
¡
∈ L1 (P ) since we assume in (2.13) finiteness of the maximal expected
also U (τn0 , Yn )
utility. The same arguments apply to U (τ 0 , Yn ).
Let us first argue (2.16). For ease of notation, we drop in the rest of this step all the 0
since the presence or absence of the minimum with σk does not change the argument in any
¢
¡
way. We first fix m and note that U (τn , Yn )I{ m1 ≤Yn ≤m} = U τn , Yn I{ m1 ≤Yn ≤m} I{ m1 ≤Yn ≤m}
(with the usual convention that ±∞ · 0 = 0). If |U | denotes the variation of U with respect
¯
¯
1
) and therefore, using the
to t, we thus have ¯U (τn , Yn )I{ m1 ≤Yn ≤m} ¯ ≤ |U |(τn , m) + |U |(τn , m
assumption (2.11),
¯
¯
sup ¯U (τn , Yn )I{ m1 ≤Yn ≤m} I{τn <%m } ¯ ∈ L1 (P )
(2.18)
n∈IN
for a localising sequence (%m ), constructed via a diagonal argument, which makes each
1
) have integrable variation. Moreover, U (τn , Yn ) ∈ L1 (P ) implies that
|U |(·, m) and |U |(·, m
U (τn , 0)I{Yn =0} ∈ L1 (P ) so that
(2.19)
U (τn , Yn )I{Yn =0} = U (τn , 0)I{U (τn ,0)>−∞} I{Yn =0}
P -a.s.
¢
¡
Hence we also have from (2.11) possibly after further modifying (%m ) that
¯
¯
(2.20)
sup ¯U (τn , 0)I{U (τn ,0)>−∞} I{Yn =0} I{τn <%m } ¯ ∈ L1 (P ).
n∈IN
Clearly, (2.18) and (2.20) also hold with τn replaced by τ everywhere.
Now introduce for m, n ∈ IN the double sequence of nonnegative random variables
¡
¢
Zm,n := |U (τn , Yn ) − U (τ, Yn )| I{ m1 ≤Yn ≤m} + I{Yn =0} I{τn <%m } .
Then for each m ∈ IN , (2.12) and (2.19) combined with (2.11) imply that lim Zm,n = 0
n→∞
1
P -a.s., and since sup Zm,n ∈ L (P ) by (2.18) and (2.20), we get
n∈IN
for each m ∈ IN
lim E[Zm,n ] = 0
n→∞
18
by dominated convergence. On the other hand, for each n ∈ IN , the sequence (Zm,n )m∈IN
increases to Z∞,n := |U (τn , Yn ) − U (τ, Yn )| P -a.s., and so monotone integration yields
lim E[Zm,n ] = E[Z∞,n ] = E[|U (τn , Yn ) − U (τ, Yn )|]
for each m ∈ IN .
m→∞
But now we use the general fact that lim sup lim inf amn ≤ lim inf lim sup amn for any double
m→∞
n→∞
m→∞
n→∞
sequence (amn ) in IR to obtain
lim sup E[|U (τn , Yn ) − U (τ, Yn )|] = lim sup lim E[Zm,n ] ≤ lim inf lim E[Zm,n ] = 0.
n→∞ m→∞
n→∞
m→∞ n→∞
This proves (2.16). Finally, (2.17) is proved completely analogously by working with the
¢
¡
q.e.d.
random variables Z̄m,n := |U (τ, Yn ) − U (τ, Y∞ )| I{ m1 ≤Yn ≤m} + I{Yn =0} I{τ <%m } .
Remark 2.9. As can be seen from the above proof, the conclusion of Theorem 2.8 still holds
if we replace (2.9) by the slightly weaker assumption that De,σ (S, P ) 6= ∅. In particular, by
the fundamental theorem of asset pricing, the latter holds if S satisfies (NFLVR).
¦
b with nice trajectories. In order to apply
The next logical step is to obtain a version of X
b satisfies (NUPBR) for simple admissible integrands for
Lemma 2.1, we want to argue that X
b To that end, we first prove that simple admissible integrals of X
b can (modulo suitable
X.
stopping) be realised via (general) admissible integrals of S. More precisely:
Proposition 2.10. Suppose that the IRd -valued semimartingale S satisfies (2.9), and U
satisfies (2.13). For any fixed T ∈ (0, ∞), there exists a sequence (Tm )m∈IN of stopping times
increasing stationarily to T and having the following property: Let H be a simple real-valued
I
P
hi I]]τi−1 ,τi ]] , where 0 ≤ τ0 ≤ τ1 ≤ · · · ≤ τI ≤ T are stopping
integrand of the form H =
i=1
times and hi is a bounded Fτi−1 -measurable random variable for i = 1, . . . , I. Fix x > 0 and
b ≥ 0, then there exist x-admissible integrands
take any stopping time % ≤ T . If x + H . X
ϑ(m,%) , m ∈ IN , for S Tm such that for each m,
(m,%)
b%∧T = x + ϑ(m,%) . S%∧T = X x,ϑ
x + H .X
m
m
%∧Tm
P -a.s.
Proof. Take W := W (T ) from Lemma 2.5 and apply Proposition 2.7 to obtain stopping times
(Tm )m∈IN increasing stationarily to T and probability measures Qm , m ∈ IN , equivalent to
P such that for each m, the random variable W is in L2 (Qm ) and S Tm is in M2 (Qm ).
1) Suppose first that each hi is not a random variable, but a constant ci ∈ IR. Then we
have for any stopping time τ ≤ T that
bτ =
H .X
I
X
i=1
bτ ∧τ − X
bτ ∧τ ) =
ci ( X
i
i−1
I
X
i=1
ci
Zτ
0
¡
19
¢
ϑ̂(τi ∧ τ, u) − ϑ̂(τi−1 ∧ τ, u) dSu = ψ (τ ) . Sτ
(τ )
with the predictable process ψu :=
I
P
i=1
¡
¢
ci ϑ̂(τi ∧τ, u)− ϑ̂(τi−1 ∧τ, u) . Since S Tm ∈ M2 (Qm )
:= ψ (%∧Tm ) yields that the process ϑ(m,%) . S Tm = ϑ(m,%) . S
and W ∈ L (Qm ), choosing ϑ
is a martingale in M2 (Qm ), and by construction,
2
(m,%)
b%∧T ≥ 0
x + ϑ(m,%) . S%∧Tm = x + H . X
m
P -a.s.
Thus also x + ϑ(m,%) . S ≥ 0 by the martingale property, and so ϑ(m,%) is x-admissible for S Tm .
2) In general, if hi is an Fτi−1 -measurable random variable, we can no longer simply say
¢
¡
that hi ϑ̂(τi ∧ τ, ·) − ϑ̂(τi−1 ∧ τ, ·) is predictable. Therefore we define
(N )
hi
(N )
Thus hi
:= −N I{hi <−N } + N I{hi ≥N } +
NX
2N −1
k=−N 2N
k2−N I{k2−N ≤hi <(k+1)2−N } .
(N )
(N )
is a finite sum of terms of the form ck IA(N ) with sets Ak
k
(N )
(i)
∈ Fτi−1 and constants
(i)
ck . So if we define stopping times %k ≤ τi by %k := τi−1 IA(N ) + τi I(A(N ) )c , we obtain
k
(N )
(N )
ck IA(N ) I]]τi−1 ,τi ]] = ck I]]%(i) ,τi ]] , and so H (N ) :=
k
k
I
P
i=1
(N )
hi
k
I]]τi−1 ,τi ]] has the same form as in
step 1) (although it now involves a double sum over i and k). Hence we get as in step 1)
predictable integrands ϑ(N,m,%) for S Tm with
b%∧T ,
x + ϑ(N,m,%) . S%∧Tm = x + H (N ) . X
m
(N )
m
and each ϑ(N,m,%) . S T is in M2 (Qm ). But clearly each hi
N → ∞, and therefore also
b%∧T =
H (N ) . X
m
I
X
i=1
(N ) ¡
hi
converges to hi pointwise as
¢
bτ ∧%∧T − X
bτ ∧%∧T −→ H . X
b%∧T
X
i
m
i−1
m
m
P -a.s. as N → ∞.
¢
¡
b%∧T
Moreover, the sequence H (N ) . X
m N ∈IN has the majorant
¯
¯
¡
¢
bτ ∧%∧T + X
b%∧T ¯ ≤ IkHk∞ max X
bτ ∧%∧T ≤ 2IkHk∞ W (T ) ∈ L2 (Qm ),
sup ¯H (N ) . X
m
i
m
i−1
m
N ∈IN
i=1,...,I
b%∧T also converges in L2 (Qm ) as N → ∞. But the
and thus ϑ(N,m,%) . S%∧Tm = H (N ) . X
m
construction of stochastic integrals with respect to martingales now implies that there exists
an S Tm -integrable process ϑ(m,%) such that ϑ(m,%) . S Tm is in M2 (Qm ) and has final value
b%∧T ≥ −x. The martingale property of ϑ(m,%) . S Tm therefore implies
ϑ(m,%) . S%∧Tm = H . X
m
that ϑ(m,%) is x-admissible for S Tm , and so the proof is complete.
20
q.e.d.
Proposition 2.11. Suppose that the IRd -valued semimartingale S satisfies (2.9), and U
b satisfies (NUPBR) for simple x-admissible integrands. As a consesatisfies (2.13). Then X
b admits P -a.s. right and left limits along the rationals.
quence, X
b fails (NUPBR) for simple
Proof. We fix T ∈ [0, ∞] and argue by contradiction. If X
b
x-admissible integrands, we can find a sequence of simple predictable integrands H n for X
b ≥ −x and (H n . X
bT )n∈IN is not bounded in L0 . By Proposition 2.10, we
such that H n . X
bT = ϑ(m,n) . ST with ϑ(m,n) . S ≥ −x on [[0, Tm ]] since H n . X
b ≥ −x. If we
can write H n . X
m
m
extend ϑ(m,n) to [[0, T ]] by setting it to 0 after Tm , we thus get ϑ(m,n) . S ≥ −x on [[0, T ]] so
that (ϑ(m,n) )n∈IN is for each m a sequence of x-admissible integrands for S. But S satisfies
(2.4) by Lemma 2.4, and so for each m,
(2.21)
Now
¢
¡
the sequence ϑ(m,n) . ST n∈IN is bounded in L0 .
P [ϑ(m,n) . ST ≥ η] ≥ P [ϑ(m,n) . ST ≥ η, T = Tm ]
bT ≥ η, T = Tm ]
= P [H n . X
bT ≥ η] − P [Tm < T ],
≥ P [H n . X
and since Tm % T , we have for any δ > 0 that P [Tm < T ] ≤ 2δ for m large enough. So if
bT )n∈IN is not bounded in L0 , we have lim sup P [H n . X
bT ≥ η] ≥ δ for some δ > 0 and
(H n . X
n→∞
all η > 0, and hence for m fixed large enough also lim sup P [ϑ(m,n) . ST ≥ η] ≥
n→∞
δ
2
for all η > 0,
b indeed satisfies (NUPBR) for simple x-admissible
which contradicts (2.21). Therefore X
integrands, and the existence of right and left limits along the rationals then follows from
Lemma 2.1.
q.e.d.
Remark 2.12. 1) By slightly modifying the proof of Proposition 2.11, we could also show
b satisfies (NFLVR)
that if S satisfies (NFLVR) and (2.9), and U still satisfies (2.13), then X
for simple admissible integrands. This gives again existence of right and left limits along the
rationals, by Lemma 2.1.
2) For the case where U does not depend on (ω, t), an alternative way to obtain the
conclusion of Proposition 2.11 would be to invoke Proposition 7.2.2 of Ankirchner (2005).
That result shows that an adapted RCLL process Y satisfies (NFLVR) for simple strategies if
its expected utility remains bounded over all final wealths from simple strategies, and a closer
look at the proof reveals that the assumption of RCLL trajectories is not needed. Since we
b as general stochastic integrals of S via Proposition
can rewrite simple stochastic integrals of X
2.10 while preserving admissibility, we could thus invoke (2.13) to apply Ankirchner’s result.
However, our result allows more general U and our proof is also more direct.
21
¦
Proposition 2.13. Suppose that the IRd -valued semimartingale S satisfies (2.9), and U
bτ
b admits a version X
e which is RCLL, and for which X
eτ = X
satisfies (2.11)–(2.13). Then X
e = H .X
b for every simple
P -a.s. for every bounded stopping time τ . As a consequence, H . X
predictable process H.
et :=
Proof. As in Lemma 2.2, set X
lim
u&t, u∈I
Q
bu for t ≥ 0; this is well defined by Proposition
X
bτ P -a.s. for
eτ = X
2.11, and Theorem 2.8 implies like in (the proof of) Lemma 2.2 that X
e is clearly a version of X.
b Moreover, H . X
e = H .X
b
each bounded stopping time τ . Hence X
e satisfies
for every simple predictable process H, and so Proposition 2.11 implies that also X
e But by construction, X
e is adapted, like
(NUPBR) for simple x-admissible integrands for X.
b and right-continuous, hence optional and so progressively measurable, and thus X
e also
X,
admits left limits by the second part of Lemma 2.1. This completes the proof.
q.e.d.
e from Proposition 2.13 is a semimartingale. The basic
Our goal is now to show that X
idea for that is similar to the approach taken by S. Ankirchner and P. Imkeller in Ankirchner
(2005) and Ankirchner/Imkeller (2005), and goes back to the classical ideas in Delbaen/
Schachermayer (1994). Ankirchner and Imkeller deduce the semimartingale property from
the finiteness of the maximal expected utility from final wealth when this is taken over all
simple strategies; but all that is really needed is Lemma 2.3 which says in essence that a
locally bounded process Y is a semimartingale if the family of final wealths from bounded
simple admissible strategies for Y is bounded in L0 . We then argue indirectly to show that S
e admits a sequence of simple admissible strategies with final wealths unbounded
fails (2.4) if X
in L0 ; this can be achieved because Proposition 2.10 allows us to translate simple admissible
e into admissible integrals over S.
integrals over X
e and to be able to use the general results in
In order to deal with the jumps of X
e
Section 2.1 on locally bounded processes, we introduce the process V of large jumps of X
P
es I
e
∆X
by Vt :=
es |>1} for t ≥ 0. This is well defined, since X is RCLL, and clearly
{|∆X
0<s≤t
e is a semimartingale
adapted, RCLL and of finite variation, hence a semimartingale. So X
e − V is a semimartingale, and clearly X̄ is locally bounded since its
if and only if X̄ := X
jumps are bounded by 1, by construction. To prove that X̄ is a semimartingale, it is enough
to prove that X̄ T is a semimartingale for each T ∈ (0, ∞).
Now fix x > 0 and T ∈ (0, ∞) and construct the stopping times (Tm )m∈IN as in Proposition 2.7 with W = W (T ) taken from Lemma 2.5. Set
and define now
(2.22)
¯
©
σm := inf t ≥ 0 ¯ |V |t ≥
m
2
et >
or X
%m := Tm ∧ σm ∧ T.
22
m
2
ª
So (%m )m∈IN increases to T stationarily, and if each X̄ %m is a semimartingale, so is X̄ T .
We are now ready to prove the main result of this section.
Theorem 2.14. Suppose that the IRd -valued semimartingale S satisfies (2.9), and U satb of optimal wealths has a version X
e which is a
isfies (2.11)–(2.13). Then the process X
semimartingale (and RCLL).
e of X
b in Proposition 2.13, and
Proof. 1) We have already constructed an RCLL version X
e is a semimartingale if and only if the locally bounded process
we have seen above that X
et −
X̄t := X
X
0<s≤t
es I
∆X
e
{|∆X
s |>1}
e t − Vt ,
=X
t≥0
is a semimartingale, or equivalently if and only if X̄ %m is a semimartingale for each m, where
%m is constructed in (2.22). So fix m and suppose that X̄ %m is not a semimartingale. We
want to show that this leads to a contradiction to (2.9). For brevity, write X := X̄ %m and
note that all stochastic integrals of X end at time T at the latest. Since X is adapted and
locally bounded but not a semimartingale, Lemma 2.3 gives a sequence of simple predictable
processes |H n | ≤ 1 with H n . X ≥ −c for all n, for some c > 0, and
(2.23)
P [H n . X%m ≥ αn ] = P [H n . XT ≥ αn ] ≥ δ > 0
for all n
with a sequence αn → ∞ and some δ > 0, because (H n . XT )n∈IN is not bounded in L0 . So
we have a sequence of stochastic integrals of X whose final values are not bounded in L0 ; but
e whose final values
what we really want is a sequence of stochastic integrals of the process X
are not bounded in L0 . This will be constructed next.
e ≥ −x with X
e0 = x and note from the construction of %m ≤ σm that
2) Recall that X
m
n
e
|V |t ≤ m
2 and Xt ≤ 2 for t < %m . Because H ≥ −1, we thus have for t < %m that
et − X
e0 ≥ −c−|V |t −x− X
e0 ≥ −c− m −2x ≥ −c− 3m −4x.
et = H n . Xt +H n . Vt + X
(H n +1) . X
2
2
Using H n + 1 ≤ 2, we obtain at time %m that
e% = (H n + 1) . X
e% − + (H n + 1)X
e% − (H%n + 1)X
e% −
(H n + 1) . X
%m
m
m
m
m
m
≥ (−c −
m
2
− 2x) − 2x − 2 m
2 = −c −
so that
(2.24)
e %m ≥ −c −
(H n + 1) . X
23
3m
2
− 4x.
3m
2
− 4x
e we note that
To exploit (2.23) in terms of X,
e −X
e + 2x + |V |
e +X
e0 − H n . V ≤ (H n + 1) . X
H n . X = (H n + 1) . X
e − x ≥ −2x. Therefore
e −X
e0 = X
because X
e% ≥ αn − k] + P [2x + |V |% ≥ k],
P [H n . X%m ≥ αn ] ≤ P [(H n + 1) . X
m
m
for any fixed k, and if we choose k large enough, we can ensure that P [2x + |V |%m ≥ k] ≤
where δ > 0 comes from (2.23). Thus we get, for fixed m and k large but fixed,
e% ≥ αn − k] ≥
P [(H n + 1) . X
m
(2.25)
in view of (2.23).
e n :=
Now keep m and k fixed and define H
simple predictable processes. Then (2.24) gives
e %m =
e n .X
H
(2.26)
e n is
so that each H
x
2 -admissible,
2(c +
+ 4x)
for all n ∈ IN
x
n
2c+3m+8x (H
(αn −k)x
2c+3m+8x ,
+ 1) for n ∈ IN , which are
e %m ≥ − x
(H n + 1) . X
2
e %m , and
hence x-admissible for X
£
e% ≥ z] = P (H n + 1) . X
e n .X
e% ≥
P [H
m
m
shows that for βn :=
(2.27)
x
3m
2
δ
2
δ
2
z(2c+3m+8x) ¤
x
we have
e % ≥ βn ] ≥
e n .X
P [H
m
δ
2
for all n ∈ IN
from (2.25). Moreover, αn → ∞ implies that βn → ∞ since x, c, m, k are all fixed. In
e n )n∈IN of simple predictable processes which
summary, then, we have found a sequence (H
e have final values which
e %m and whose integrals with respect to X
are all x-admissible for X
are not bounded in L0 because they satisfy (2.27) with some δ > 0 and a sequence βn → ∞.
bτ for every bounded stopping time τ
eτ = H . X
3) Recall from Proposition 2.13 that H . X
and every simple predictable process H. For every n ∈ IN , we thus obtain from Proposition
2.10 an x-admissible integrand ϑ(n,m,%m ) for S Tm such that
e n .X
e% = x + H
b% = x + ϑ(n,m,%m ) . S%
e n .X
x+H
m
m
m
P -a.s.;
¢
¡
note that %m ≤ Tm by (2.22). Therefore we have a sequence x + ϑ(n,m,%m ) . S%m n∈IN in X%xm
which is not bounded. But this contradicts (2.4) and hence also (2.9) by Lemma 2.4, and so
X̄ %
m
must be a semimartingale for each m. Thus the proof is complete.
24
q.e.d.
Remark 2.15. There is a whole strand of literature with results of the form that, loosely
speaking, a process Y is a semimartingale if the maximal expected “utility” from its stochastic integrals over simple integrands remains finite; see Chapter 7 of Ankirchner (2005),
Ankirchner/Imkeller (2005), Larsen/Žitković (2008), or Biagini/Øksendal (2005) in a slightly
different setting. However, these results typically assume that Y is locally bounded and that
the increasing “utility” function U : [0, ∞) → IR satisfies lim U (y) = +∞. Since our utility
y→∞
e is not locally bounded,
U might be bounded and can depend on (ω, t) as well, and since X
a translation to our setting is not straightforward. The reason why we can still obtain the
e is that because X
e comes from optimal investment, its simple
semimartingale property for X
integrals can be related (via Proposition 2.10) to integrals of S. In that sense, we exploit
e has more structure than some abstract process Y .
that X
¦
Our goal in Section 5 is to provide more details on the structure of the semimartingale
e In preparation, we study in the next section the structure of the optimal strategy field ϑ̂
X.
in more detail.
3. Properties of the optimal strategy field
¡
¢
In this section, we study the properties of the two-parameter process ϑ̂(t, u) t≥u≥0 in more
detail. Recall that ϑ̂(t, ·) is an optimal strategy for the utility maximisation problem with
time horizon t. The properties of (a good version ϑe of) ϑ̂ will be crucial in the sequel when
e of optimal wealths from
we want to establish the precise structure for the semimartingale X
Theorem 2.14. In a nutshell, our goal in this section is to prove that ϑ̂ admits a version ϑe
such that
e u) is of finite variation, and
– as a function of the time horizon, t 7→ ϑ(t,
e t), t ≥ 0, is predictable and S-integrable.
– the diagonal ϑ̄t = ϑ(t,
For technical reasons, however, a precise statement is a bit more involved.
To the best of our knowledge, there exist no results in the literature that are even
remotely close to the above precise structural description of horizon-dependence in optimal
portfolio choice. In general terms, financial economists are of course well aware that decision
making under uncertainty is strongly affected by the planning horizon. But the considerable
literature on that topic is almost exclusively empirical and aims predominantly at a qualitative
understanding of which factors have which kind of influence, etc.; see Siebenmorgen/Weber
(2004) for a (randomly chosen) recent sample of that literature. The only work we have
found giving any theoretical result is an unpublished preprint by A. Lazrak which presents
a continuous-time version of an idea from (an early preprint version of) Gollier/Zeckhauser
25
(2002). In the very specific setting of Lazrak (1998), S is a multivariate geometric Brownian
motion with a unique equivalent martingale measure, and y 7→ U (y) is a standard utility
function on (0, ∞). The optimal strategy can then be computed in almost closed form,
and the (from our perspective) main result in Lazrak (1998) more or less shows that this
strategy is increasing with respect to the time horizon if and only if the risk tolerance function
y 7→ −U 0 (y)/U 00 (y) of U is convex. But it is clear from the arguments that this result and
its proof depend very strongly on the specific model in Lazrak (1998).
On the quantitative or mathematical side, there is also a substantial literature on the
asymptotic behaviour of optimal strategies as the time horizon T tends to ∞. This can
be found under headings like risk-sensitive control/portfolio management or portfolio turn-
pike theorems. In a different direction, several authors have tried to find conditions or approaches that eliminate horizon-dependence by a modified problem formulation; see for instance Henderson/Hobson (2007), Choulli/Stricker/Li (2007), Zariphopoulou/Žitković (2008)
or the numerous papers by M. Musiela and T. Zariphopoulou on forward preference criteria.
But neither the results nor the techniques in all these areas seem to help in obtaining more
precise information on the structure of horizon-dependence for all time horizons T ∈ (0, ∞).
We do believe that our study is the first to address and answer such questions in a general
framework in a rigorous way.
After these general remarks, let us now get started. Throughout this section, we
assume that the IRd -valued semimartingale S satisfies (2.9), and U satisfies (2.11)–
(2.13), so that we can use all the results from Section 2. In order to have good properties
for several quantities appearing below, we need the following preliminary result. Recall that
the set WT is defined in Lemma 2.5.
Lemma 3.1. Assume that the IRd -valued semimartingale S satisfies (2.9). For every fixed
T ∈ (0, ∞), there exist stopping times (Tm )m∈IN increasing stationarily to T and probability
measures (Qm )m∈IN equivalent to P with the following properties for each m:
(3.1) W (T ) = ess sup WT ∈ L2 (Qm ).
e ∗ = sup X
et ∈ L2 (Qm ).
(3.2) X
T
0≤t≤T
(3.3) J
(3.4) J
(1)
(2)
¯
o
¯
∗
.
.
:= ess sup j ST = sup |j St | ¯ j predictable with kjk∞ ≤ 1 ∈ L2 (Qm ).
n
:= ess sup
(Πn )n∈IN
½
0≤t≤T
sup
P
n∈IN τi ∈Πn
eτ − X
eτ )
(X
i+1
i
2
¾
∈ L2 (Qm ), where the essential supremum
is taken over all sequences (Πn )n∈IN of random partitions (by stopping times τi ) of
[0, T ] whose mesh size |Πn | := sup{τi − τi−1 | τi−1 , τi ∈ Πn } tends to 0.
(3.5) S Tm ∈ M2 (Qm ).
e Tm = X
e0 + M̃ m + Ãm with M̃ m ∈ M2 (Qm ) and Ãm predictable with uniformly
(3.6) X
0
26
bounded variation |Ãm |.
(3.7) [M̃ m , Ãm ] is a Qm -martingale.
R
e− dM̃ m is a Qm -martingale.
(3.8)
X
R ∗
e− d|Ãm | + hM̃ m i + [Ãm ] is uniformly bounded.
(3.9) B̃ m := 4 X
Proof. By Lemma 2.5, the random variable W (T ) is finite P -a.s. So is J (1) because S
e is a semimartingale by Theorem 2.14 so that for any
is a semimartingale. Moreover, X
sequence (Πn )n∈IN of random partitions of [0, T ] whose mesh size |Πn | tends to 0, the sequence
´
³ P
e τ )2
eτ − X
e T in L0 ; see Theorem II.22 in Protter (2005).
(X
converges to [X]
i+1
i
τi ∈Πn
n∈IN
Thus also J (2) is finite P -a.s. as the essential supremum of a family which is bounded in L0 .
We now apply Proposition 2.7 with the random variable W := W (T ) + J (1) + J (2) to ob0
tain (Tm
) and (Qm ) which already satisfy (3.1)–(3.4); note that (3.2) follows by observing that
0
0
e Tm
e ∗ ≤ W (T ) . Moreover, S Tm
∈ M2 (Qm ) and X
is a semimartingale which is Qm -special
X
T
0
e0 + M̃ + Ã under
e Tm
since its supremum is in L2 (Qm ). For its canonical decomposition X
=X
Qm , we can then obtain (3.6)–(3.9) as well as (3.5) by stopping; note that [M̃ , Ã] is a local
Qm -martingale by Yoeurp’s lemma (see Dellacherie/Meyer (1982), Theorem VII.36), and |Ã|
R ∗
e− d|Ã| + hM̃ i + [Ã] are predictable and RCLL, hence locally bounded. We
and B̃ := 4 X
0
and the stopping times required for the above localisations,
choose as Tm the minimum of Tm
and the proof is complete.
q.e.d.
Now we introduce a suitably “localised version” of ϑ̂ for which we can subsequently
establish precise results. Choose first an arbitrary T ∈ (0, ∞) and construct Tm and Qm as
in Lemma 3.1. Then we fix m ∈ IN and define the process ϑ(m) by
ϑ(m) (t, u) := ϑ̂(t ∧ Tm , u ∧ Tm )
for 0 ≤ t ≤ T , 0 ≤ u ≤ T .
Note that ϑ(m) (t, u) = 0 for u ∧ Tm > t ∧ Tm since ϑ̂(t ∧ Tm , ·), being a strategy on [[0, t ∧ Tm ]],
is only defined on that interval. Recall that Ω̄ := Ω × (0, ∞) and set F̄ := F ⊗ B(0, ∞).
Since S is IRd -valued, we need a bit of notation in connection with our stochastic integrals. The main issue is that the bracket process ([S] or hSi) of S is matrix-valued. So
we choose a bounded (real-valued) predictable increasing process B m null at 0 such that
®
­ T i
(S m ) , (S Tm )j ¿ B m for all i, j = 1, . . . , d (where these brackets are taken under Qm ); one
³P
d ­
®´
example would be B m = tanh
(S Tm )i . We introduce the matrix-valued predictable
i=1
®±
­ T i
m
m
m
process σ by σij = d (S ) , (S Tm )j dB m and the finite measure µm := Qm ⊗ B m on F̄.
Then we have the isometry
(3.10)
kϑ . S Tm kM2 (Qm ) = kϑ . STTm kL2 (Qm )
27
°
1°
= °(ϑtr σ m ϑ) 2 °L2 (µ
m)
1
2
= kϑtr σ m ϑkL1 (µm )
=: kϑkL2 (Qm ,S Tm ) .
Moreover, we also have
­
(3.11)
ϑ.S
Tm
, ψ .S
Tm
®
=
Z
ϑtr σψ dB m .
Finally, we note that by its definition, each σ m (u) is symmetric and nonnegative definite;
hence σ m = (cm )tr cm for some matrix-valued process cm , and we can and do choose cm to
be predictable as well.
The properties (3.10) and (3.11) are precisely what we need to derive properties for ϑ(m)
e Tm . We remark that for the case d = 1 where S is real-valued, we can directly
from those of X
take B m = hS Tm i so that then µm = Qm ⊗ hS Tm i and σ m ≡ 1.
Lemma 3.2. Viewed as a process defined on Ω̄, the mapping t 7→ ϑ(m) (t, ·) is right-continuous in L2 (Qm , S Tm ). Equivalently, the mapping t 7→ cm ϑ(m) (t, ·) is right-continuous in
L2 (µm ) and hence in (P ⊗ B m )-measure.
Proof. Take t0 > t and write
(3.12)
et∧T = X
bt0 ∧T − X
bt∧T
et0 ∧T − X
X
m
m
m
m
=
t0Z
∧Tm
ϑ̂(t0 ∧ Tm , u) dSu −
0
=
ZT
0
=
ZT
0
t∧T
Z m
0
ϑ̂(t ∧ Tm , u) dSu
¡
¢
ϑ̂(t0 ∧ Tm , u) − ϑ̂(t ∧ Tm , u) dSuTm
¡
¢
ϑ̂(t0 ∧ Tm , u ∧ Tm ) − ϑ̂(t ∧ Tm , u ∧ Tm ) dSuTm
by using that ϑ̂ vanishes whenever its second argument exceeds the first one. Since all the
ϑ̂(τ, ·) are x-admissible, all stochastic integrals are local Qm -martingales due to (2.7), and
they are even in M2 (Qm ) because W (T ) is a majorant for their suprema which is in L2 (Qm )
by (3.1) in Lemma 3.1. Hence we obtain
(3.13)
£
et∧T )
et0 ∧T − X
EQ m ( X
m
m
¤
2
= EQ m
"µ ZT
0
¡
¢
ϑ(m) (t0 , u) − ϑ(m) (t, u) dSuTm
°2
°
= °ϑ(m) (t0 , ·) − ϑ(m) (t, ·)° 2
L (Qm ,S Tm )
28
.
¶2 #
et∧T converges to 0 P -a.s. as t0 & t by right-continuity of X
et0 ∧T − X
e (or in L0 by
But X
m
m
¡ (T ) ¢2
, so that
Theorem 2.8), and a Qm -integrable majorant for the squares is given by W
¤
£
et∧T )2 = 0 by dominated convergence. This gives the assertion thanks
et0 ∧T − X
lim EQ (X
t0 &t
m
m
m
to the isometry (3.10).
q.e.d.
By pushing the same idea a little further, we next establish a measurability result for ϑ̂,
or rather ϑ(m) . Recall that T ∈ (0, ∞) is fixed but arbitrary.
Lemma 3.3. Viewed as a mapping on [0, T ] × Ω̄, ϑ(m) has a version which is (B[0, T ] ⊗ P)¯
measurable, where “version” is with respect to the measure λ¯[0,T ] ⊗ µm on [0, T ] × Ω̄.
Proof. Denote by Dk the k-th dyadic partition of [0, T ] and by d0 := d + T 2−k the successor
P
in Dk of d ∈ Dk . Define then g k (t, ·) :=
I(d,d0 ] (t)ϑ(m) (d0 , ·) and note that each g k is
d∈Dk
clearly (B[0, T ] ⊗ P)-measurable because each ϑ(m) (d0 , ·) is a predictable process. Moreover,
for each fixed t, we have cm g k (t, ·) → cm ϑ(m) (t, ·) in L2 (µm ) as k → ∞ by Lemma 3.2; but
we can do even better. Indeed, write first
¯
¡
¢¯2
(3.14) ¯cm (u) g k (t, u) − ϑ(m) (t, u) ¯
¡
¢tr
¢
¡
= g k (t, u) − ϑ(m) (t, u) σ m (u) g k (t, u) − ϑ(m) (t, u)
µ X
¶
¡ (m) 0
¢ tr m
(m)
I(d,d0 ] (t) ϑ (d , u) − ϑ (t, u)
=
σ (u)
d∈Dk
×
=
µ X
I(d,d0 ] (t) ϑ
d∈Dk
X
¡
I(d,d0 ] (t)
d∈Dk
³¡
ϑ
(m)
(m)
0
(d , u) − ϑ
0
(d , u) − ϑ
(m)
(m)
(t, u)
(t, u)
¢
¢tr
¶
¡
m
σ (u) ϑ
(m)
0
(d , u) − ϑ
(m)
(t, u)
¢´
;
then integrate with respect to B m in u, take expectations under Qm in ω, and finally integrate
with respect to λ in t to obtain, by using (3.10) and (3.13),
ZT °
¢¯ °
° ¯¯ m ¡ k
°2
(m)
° c g (t, ·) − ϑ (t, ·) ¯ ° 2
L (µm )
dt
0
=
ZT
0
=
° k
°
°g (t, ·) − ϑ(m) (t, ·)°2 2
ZT X
0 d∈Dk
L (Qm ,S Tm )
dt
°
°2
I(d,d0 ] (t)°ϑ(m) (d0 , ·) − ϑ(m) (t, ·)°L2 (Qm ,S Tm ) dt
29
=
ZT X
0 d∈Dk
=
ZT
0
=
ZT
0
if we set Ytk :=
P
d∈Dk
EQ m
£
¤
ed0 ∧T − X
et∧T )2 dt
I(d,d0 ] (t)EQm (X
m
m
"µ
X
d∈Dk
ed0 ∧T − X
et∧T )
I(d,d0 ] (t)(X
m
m
¶2 #
dt
£
¤
etTm )2 dt,
EQm (Ytk − X
e T0m for 0 ≤ t ≤ T . For every fixed t, Ytk converges to X
etTm
I(d,d0 ] (t)X
d
e (or in L0 by Theorem 2.8), and
P -a.s. as k → ∞ by the right-continuity of X
etTm | ≤ W (T ) ∈ L2 (Qm ).
sup sup |Ytk | ≤ sup |X
k∈IN 0≤t≤T
0≤t≤T
£
¤
etTm )2 tends to 0 as
Hence dominated convergence first yields that fk (t) := EQm (Ytk − X
£
¤
k → ∞ for each t; but since we also have fk (t) ≤ 4E (W (T ) )2 for all t and k, again using
dominated convergence gives lim
RT
k→∞ 0
fk (t) dt = 0. Translated back, this means that
|cm g k − cm ϑ(m) | −→ 0
¡ ¯
¢
in L2 λ¯[0,T ] ⊗ µm as k → ∞,
so that cm ϑ(m) agrees almost everywhere on [0, T ] × Ω̄ with lim sup cm g k =: hm , say, which
k→∞
m
k
is (B[0, T ] ⊗ P)-measurable since c and each g are. Finally, we write hm = cm ψIC + 0IC c
for a set C ∈ B[0, T ] ⊗ P and a (B[0, T ] ⊗ P)-measurable process ψ, by distinguishing in a
¢
¢
¡
¡
(B[0, T ]⊗P)-measurable way whether hm (t, u) is in Range cm (u) or in Ker cm (u) . Setting
then ϑ(m) := ψ gives the desired version.
q.e.d.
Remark 3.4. Although, as pointed out in Section 2 after (2.13), we have no uniqueness for
the optimal strategy ϑ̂(τ, ·) for a given time horizon τ , Lemma 3.3 nevertheless allows us to
choose a good version for ϑ(m) . Once we have done that, the nonuniqueness issue is laid to
¦
rest from here on.
In the sequel, we choose and fix a version of ϑ(m) as in Lemma 3.3, without
changing the notation. For future use, we note that this has no effect on integrals formed
of simple integrands in the t-variable with respect to ϑ(m) , since the resulting integrals are
then also versions of each other.
¡
¢
Lemma 3.5. Suppose that the process Y (t, u) t≥0, u≥0 is (B[0, ∞) ⊗ P)-measurable when
viewed as a mapping on [0, ∞) × Ω̄. Then the diagonal process Ȳt := Y (t, t), t ≥ 0, is
predictable. (An analogous result holds true if P is replaced by the optional σ-field O.)
30
Proof. If Y (t, u) = f (t)gu where g is a predictable process (on Ω) and f is a Borel-measurable
¡
¢
function on [0, ∞) , then Ȳt = f (t)gt is predictable as the product of two predictable processes, since any nonrandom measurable function is trivially predictable. For general Y , the
assertion follows via the monotone class theorem.
q.e.d.
Now we are ready to establish the first (and simpler) main result of this section. Apart
from the localisation issue, this says that taking the diagonal of the optimal strategy field
yields a predictable S-integrable process.
Corollary 3.6. For the version of ϑ(m) obtained from Lemma 3.3, the diagonal process
(m)
ϑ̄u
:= ϑ(m) (u, u) = ϑ̂(u ∧ Tm , u ∧ Tm ), 0 ≤ u ≤ T , is predictable and S Tm -integrable.
Proof. Predictability is immediate from Lemma 3.5 and Lemma 3.3. Now we go back to
(3.12) in the proof of Lemma 3.2 and write this as
et ∧T =
et ∧T − X
X
i+1
m
i
m
Zti
0
¡
¢
ϑ(m) (ti+1 , u) − ϑ(m) (ti , u) dSuTm +
tZi+1
ϑ(m) (ti+1 , u) dSuTm .
ti
As already pointed out in the proof of Lemma 3.2, all the stochastic integral processes are in
M2 (Qm ), and so the two summands on the right-hand side above are orthogonal in L2 (Qm ).
P
I(d,d0 ] (u)ϑ(m) (d0 , u), where Dk is again the k-th dyadic partition
With the notation ψuk :=
d∈Dk
of [0, T ], we thus obtain from (3.10) similarly as in (3.14)
(3.15)
X
d∈Dk
£
ed0 ∧T − X
ed∧T )
EQ m ( X
m
m
¤
2
≥
X
EQ m
d∈Dk
= EQm
"µ Zd0
ϑ(m) (d0 , u) dSuTm
d
"µ
0
d
X Z ¡
ϑ(m) (d0 , u) dSuTm
d∈Dk d
¤
= EQm (ψ k . STTm )2
°
°
= °(ψ k )tr σ m ψ k °L1 (µ
£
e is a semimartingale by Theorem 2.14,
But since X
X
ed∧T )2 −→ [X
ed0 ∧T − X
e Tm ] T
(X
m
m
d∈Dk
¢
¶2 #
m)
¶2 #
.
in L0 as k → ∞,
and due to (3.4) in Lemma 3.1, dominated convergence therefore implies that the left-hand
£ T
¤
e m ]T < ∞. But for the right-hand side,
side of (3.15) converges to EQm [X
cm (u)ψuk −→ cm (u)ϑ(m) (u, u) = cm (u)ϑ̄(m)
u
31
in µm -measure as k → ∞
by Lemma 3.2, and since σ m = (cm )tr cm is nonnegative definite, Fatou’s lemma yields
° (m) tr m (m) °
°(ϑ̄ ) σ ϑ̄ °
L1 (µm )
°
°
≤ lim inf °(ψ k )tr σ m ψ k °L1 (µ )
m
k→∞
¸
· X
2
ed∧T )
ed0 ∧T − X
≤ lim EQm
(X
m
m
k→∞
d∈Dk
¤
e Tm ]T < ∞.
= EQ m [ X
£
So ϑ̄(m) is even in L2 (Qm , S Tm ) and hence S Tm -integrable.
q.e.d.
Having finished the first piece of business in this section, we now turn to showing that
ϑ̂ – or rather the chosen good version of ϑ(m) – is of finite variation with respect to its
first argument t. The overall idea is to view ϑ(m) as a process indexed by t and defined
on Ω̄ = Ω × (0, ∞), and to prove that it is a semimartingale over a constant filtration. To
make this more precise, recall that F̄ = F ⊗ B(0, ∞) and define on (Ω̄, F̄ ) the filtration
Pt := P, t ≥ 0, where P is the predictable σ-field on Ω̄. (Even more precisely, we take the
augmentation with respect to µm = Qm ⊗ B m to have the usual conditions satisfied.) Then
ϑ(m) is (Pt )-adapted since each ϑ(m) (t, ·) is a predictable process.
To use the vector process ϑ(m) as a stochastic integrator with respect to the t-variable,
we start with very simple predictable matrix-valued processes f¯ on Ω̄. Such an f¯ has the
I
P
Cui (ω)I(ti ,ti+1 ] (t) with each C i being a Pti -measurable d × d-matrix on
form f¯(t, ω, u) =
i=1
i
Ω̄ so that each C is a predictable matrix-valued process on Ω. The stochastic integral of f¯
with respect to ϑ(m) is then the vector process defined by
(m)
f¯. ϑT
:=
ZT
0
f¯t ϑ(m) (dt, ·) :=
I
X
i
¡
C ϑ
i=1
(m)
(ti+1 , ·) − ϑ
(m)
¢
(ti , ·) =:
I
X
C i ∆i+1 ϑ(m) ,
i=1
and this is an IRd -valued predictable process (indexed by u) on Ω. We want to use the classical
Bichteler–Dellacherie characterisation of semimartingales as good stochastic integrators, and
(m)
so we first study the continuity properties of the mapping f¯ 7→ f¯. ϑ .
T
Proposition 3.7. Take the version of ϑ(m) from Lemma 3.3. Then:
(m),k
1) For each coordinate k ∈ {1, . . . , d}, the mapping f 7→ f . ϑT
is continuous in
¡
£ T k ¤¢
1
on the set of bounded very simple real-valued predictable processes f on
L Qm ⊗ (S m )
¯
¤¢
¡
£
⊗ Qm ⊗ (S Tm )k .
Ω̄, equipped with the L∞ -norm with respect to the measure λ¯
[0,T ]
2) For each simple (i.e. piecewise constant) real-valued measurable function f on [0, T ],
(m)
the IRd -valued process f . ϑT is in L2 (Qm , S Tm ), and we have
(3.16)
°
°
°f . ϑ(m) °
T
L2 (Qm ,S Tm )
32
≤ Cm kf kL∞
b The notation f . ϑ(m) here should be read
with a constant Cm depending only on m and X.
T
(m)
(m)
componentwise; so f . ϑT := (f Id×d ) . ϑT .
Note that although part 2) of Proposition 3.7 will only be used in Section 4, we provide
this result for efficiency already here, since the arguments for 1) and 2) are largely parallel.
Before we can prove Proposition 3.7, we need an auxiliary result. Recall from (the proof
e0 + M̃ m + Ãm under Qm .
e Tm = X
of) Lemma 3.1 the canonical decomposition X
Lemma 3.8. With the notation ∆i+1 ϑ(m) := ϑ(m) (ti+1 , ·) − ϑ(m) (ti , ·), we have
(3.17)
EQ m
" ZT
0
(3.18)
¯
#
¯
¯
£ Tm
¤
(m)
Tm ¯
etTm ¯ Ft ∧T
et − X
∆i+1 ϑ (u) dSu ¯ Fti ∧Tm = EQm X
i
m
i+1
i
¯
¯
£
¤
m¯
= EQm Ãm
ti+1 − Ãti Fti ∧Tm
and
(3.19)
EQ m
"µ ZT
0
(3.20)
#
¶2 ¯¯
¯
£ Tm
¤
¯
etTm )2 ¯ Ft ∧T
et − X
∆i+1 ϑ(m) (u) dSuTm ¯ Fti ∧Tm = EQm (X
i
m
i+1
i
¯
¯
£
¤
≤ EQm B̃tmi+1 − B̃tmi ¯ Fti ∧Tm
for an increasing predictable uniformly bounded process B̃ m .
Proof. Purely for ease of notation, we drop in this proof all indices m and argue for Tm = T .
Moreover, we also shortly write S instead of S Tm . Define the process Y i+1 := ϑ(ti+1 , ·) . S so
that Y i+1 ∈ M2 (Q) by Proposition 2.7, and note that
et − x
=X
YTi+1 = Yti+1
i+1
i+1
because ϑ(ti+1 , ·) vanishes after ti+1 . So
ZT
0
et − X
et ,
∆i+1 ϑ(u) dSu = YTi+1 − YTi = X
i+1
i
and this yields (3.17) and (3.19). Due to (3.6) in Lemma 3.1, (3.18) then follows as well.
e Tm = x + M̃ m + Ãm to obtain
To prove (3.20), we first use Itô’s formula on X
e2 − X
et − X
et − X
e t (X
e t )2 = X
et ) = 2
e 2 − 2X
(X
ti+1
ti
i+1
i
i
i+1
i
33
tZi+1
ti
e t ) dX
e t − [X]
es + [X]
es− − X
et.
(X
i
i+1
i
e = [M̃ ] + [Ã] + 2[M̃ , Ã], and by (3.6)–(3.8) and (3.2) in Lemma 3.1, the processes
Now [X]
R
et ) dM̃ on [ti , ti+1 ] are all Q-martingales. With the usual
e− − X
[M̃ ] − hM̃ i, [M̃ , Ã], and (X
i
∗
es |, we thus get
e := sup |X
supremum notation X
t
0≤s≤t
¯
¤
£
e t )2 ¯ Ft
et − X
EQ ( X
i+1
i
i
¯
#
" tZi+1
¯
es− − X
et ) dÃs + hM̃ it − hM̃ it + [Ã]t − [Ã]t ¯¯ Ft
(X
= EQ 2
i
i+1
i
i+1
i
¯ i
ti
≤ EQ [B̃ti+1 − B̃ti | Fti ]
R ∗
e d|Ã| + hM̃ i + [Ã]. By (3.9) in Lemma 3.1, B̃ has the claimed
with the process B̃ := 4 X
−
properties.
q.e.d.
Proof of Proposition 3.7. Like in the proof of Lemma 3.8, we drop all indices m and work
with Tm = T for ease of notation.
P i
a) Suppose f¯ =
C I(t ,t ] is a very simple bounded IRd×d -valued predictable process
i
i
i+1
P i
C ∆i+1 ϑ. In addition, let the matrices C i all be diagonal so that
on Ω̄ so that f¯. ϑT =
i
i
i
σC = C σ. Take any IRd -valued predictable process h in L2 (Q, S) and write
ZT
(3.21)
0
htr
u σu
¡
XZ
ti
¢
f¯. ϑT (u) dBu =
i
i
htr
u Cu σu ∆i+1 ϑ(u) dBu
0
ti+1
X Z
i
+
htr
u Cu σu ∆i+1 ϑ(u) dBu ,
i
ti
using that ∆i+1 ϑ vanishes after time ti+1 .
We first look at the second sum where we fix i. There we use first Cauchy–Schwarz
(for the scalar product in IRd with matrix σ) pointwise for the dB-integrand, then Cauchy–
Schwarz for the dB-integrals, then conditional Cauchy–Schwarz in L2 (Q) and then the (conditional) isometry property (3.10) of the stochastic integral to obtain
¯
#
" tZi+1
¯
¯ tr i
¯
¯
¯hu Cu σu ∆i+1 ϑ(u)¯ dBu ¯ Ft
EQ
¯ i
ti
≤ EQ
≤
Ã
" tZi+1
¡
ti
i
i tr
htr
u Cu σu (Cu ) hu
¢ 21 ¡
∆i+1 ϑtr (u)σu ∆i+1 ϑ(u)
¢ 12
¯
#
¯
¯
dBu ¯ Fti
¯
¯
#! 21 Ã "µ ZT
#! 12
" tZi+1
¶2 ¯¯
¯
¯
¯
∆i+1 ϑ(u) dSu ¯ Fti
(Cui hu )tr σu (Cui hu ) dBu ¯ Fti
EQ
.
EQ
¯
¯
ti
0
34
Note that (C i )tr = C i since C i is diagonal and that we have generously estimated the integral
ti+1
R
∆i+1 ϑtr (u)σu ∆i+1 ϑ(u) dBu by integrating the same quantity from 0 to T , before using
ti
(3.10). Next we sum over i, use Cauchy–Schwarz for the sum, take expectations under Q
and apply Cauchy–Schwarz for the expectation. Then we use |C i | ≤ kf¯k∞ and the isometry
(3.10) for the resulting first factor, and (3.20) from Lemma 3.8 for the second factor, to get
(3.22)
EQ
"
#
ti+1
X Z ¯
¯
i
¯htr
¯
u Cu σu ∆i+1 ϑ(u) dBu
i
ti
Ã
≤ kf¯k∞ EQ
"
#! 21
#! 12 Ã "
ti+1
X Z
X
EQ
htr
(B̃ti+1 − B̃ti )
u σu hu dBu
i
i
ti
1
= kf¯k∞ khkL2 (Q,S) kB̃T kL2 1 (Q) .
Now we look at the first sum in (3.21), or more precisely at
P Rti ¯¯
i 0
which majorises that first sum. For each i, the process
¯
i
¯ dBu ,
htr
C
σ
∆
ϑ(u)
u
i+1
u u
¢
¡
h̃i := h sign htr C i σ∆i+1 ϑ
(3.23)
is IRd -valued, predictable, in L2 (Q, S) like h, with even |h̃i | = |h|, and it satisfies by con¯
¯
struction ¯htr C i σ∆i+1 ϑ¯ = (h̃i )tr C i σ∆i+1 ϑ. Then we set hi := h̃i I[[0,ti ]] so that clearly
(C i hi ) . ST = (C i h̃i ) . Sti
is Fti -measurable.
Moreover, (C i hi ) . S ∈ M2 (Q) since hi ∈ L2 (Q, S) and C i is bounded, and so by (3.11),
EQ
" Zti
0
#
" ZT
#
¯ tr i
¯
¯hu Cu σu ∆i+1 ϑ(u)¯ dBu = EQ
(hiu )tr Cui σu ∆i+1 ϑ(u) dBu
0
® ¤
= EQ (C i hi ) . S, (∆i+1 ϑ) . S T
£¡
¢¡
¢¤
= EQ (C i hi ) . ST (∆i+1 ϑ) . ST .
£­
But the first factor in the last expectation is Fti -measurable; so we can replace the second
one by its Fti -conditional expectation, which is given by (3.18) in Lemma 3.8, and we find
(3.24)
EQ
"
#
#
"
ti
XZ ¯
X¡
¯
¢
i
¯htr
¯
(C i hi ) . Sti (Ãti+1 − Ãti )
u Cu σu ∆i+1 ϑ(u) dBu = EQ
i
i
0
≤ EQ
35
h³
¡
max (C h ) . ST∗
i
i i
¢´
i
|Ã|T .
In summary so far, adding (3.24) and (3.22) yields in view of (3.21) that
(3.25)
EQ
" ZT
0
#
1
¯ tr ¡
¢¯
¯hu σu f¯. ϑT (u) ¯ dBu ≤ kf¯k∞ khkL2 (Q,S) kB̃T k 2 1
L (Q)
°
° 12
°
¡
¢°
.
+ ° max (C i hi ) . ST∗ °L2 (Q) °|Ã|T °∞
i
If we simply use h̃i ≡ ±h instead of (3.23), then we still get with the same arguments
¯ " ZT
#¯
¯
¯
1
¡
¢
¯
¯
2
¯
¯
.
f
ϑ
htr
σ
(u)
dB
¯ EQ
T
u ¯ ≤ kf k∞ khkL2 (Q,S) kB̃T kL1 (Q)
u u
¯
¯
(3.26)
0
°
° 12
°
¡
¢°
.
+ ° max (C i hi ) . ST∗ °L2 (Q) °|Ã|T °∞
i
At this point, we distinguish two cases:
a1) If all the C i are deterministic constant matrices ci , we can pull them out of the
stochastic integrals and estimate them above by kf¯k∞ . If in addition all the h̃i coincide with
±h, then
¡
¢
max (C i hi ) . ST∗ ≤ kf¯k∞ (h . ST∗ )
(3.27)
i
because hi . S = h̃i . S ti = ±h . S ti .
a2) If all the |h̃i | are bounded by 1, then |C i hi | ≤ kf¯k∞ and so
¡
¢
(3.28) max (C i hi ) . ST∗ ≤ kf¯k∞ ess sup{j . ST∗ | j predictable and kjk∞ ≤ 1} = kf¯k∞ J (1) .
i
In both cases, we subsequently exploit that both |Ã|T and B̃T are bounded, due to (3.6)
in Lemma 3.1 and Lemma 3.8, respectively. In case a2), we know from (3.3) in Lemma
3.1 that J (1) is in L2 (Q). In case a1), Doob’s inequality and the isometry property of the
stochastic integral yield
kh . ST∗ kL2 (Q) ≤ 2kh . ST kL2 (Q) = 2khkL2 (Q,S) .
(3.29)
b) For the proof of part 1), take now an index k ∈ {1, . . . , d} and a very simple bounded
real-valued predictable process f on Ω̄. Define f¯j` := f δjk δ`k so that the only nonzero entry
of the matrix f¯ is f¯kk = f . Then f¯. ϑT = (f . ϑk )ek , where ek is the k-th basis vector in
T
IR , and hence h := sign(f¯. ϑT ) = sign(f . ϑkT )ek , where the first sign function is applied
d
coordinatewise. With the same h̃i as in step a), we are then in case a2) from above, and so
combining (3.25) with (3.28) yields, for the above choice of h and f¯,
(3.30)
°
°
°f . ϑkT °
L1 (Q⊗[S k ])
= EQ
" ZT
0
¯
¯
¯f . ϑkT (u)¯ dhS k iu
36
#
= EQ
" ZT
0
= EQ
" ZT
¯
¯
¯f . ϑkT (u)¯σukk dBu
¯.
htr
u σu f ϑT (u) dBu
0
≤ kf k∞
³
¢
¡
#
1
2
kSTk kL2 (Q) kB̃T k∞
#
+ kJ
(1)
°
° 12 ´
°
kL2 (Q) |Ã|T °∞ ,
using that kf¯k∞ = kf k∞ and |h| = 1. This proves assertion 1).
P
ci I(ti ,ti+1 ] with constants ci is a (bounded)
c) To prove part 2), suppose that f =
i
real-valued piecewise constant function on [0, T ]. Take f¯ := f Id×d and any predictable
(IRd -valued) h ∈ L2 (Q, S). Then we get from the definition of f¯, (3.26), (3.27) and (3.29)
¯ " ZT
#¯
¯
¯ ¯¯
¯
¢
¡
¯. ϑT (u) dBu ¯¯
¯(h, f . ϑT )L2 (Q,S) ¯ = ¯EQ
htr
σ
f
u
u
¯
¯
0
³
1
° 12 ´
°
2
.
≤ kf k∞ khkL2 (Q,S) kB̃T k∞
+ 2°|Ã|T °∞
Thus h 7→ (h, f . ϑT )L2 (Q,S) is a continuous linear functional on L2 (Q, S), and this implies by
Riesz’ representation theorem that f . ϑT ∈ L2 (Q, S) with
³
1
° 21 ´
°
2
°
.
kf ϑT kL2 (Q,S) ≤ kf k∞ kB̃T k∞ + 2 |Ã|T °∞ .
This proves assertion 2) and completes the proof.
q.e.d.
Remark 3.9. 1) Once we establish that each coordinate ϑ(m),k of ϑ(m) is of finite variation
(m)
with respect to t, we can of course define (the vector process) f . ϑT for every bounded
(real-valued) measurable function f on [0, T ], and the estimate (3.16) in Proposition 3.7 then
extends to these f as well.
2) In the above proof, we have worked with the sharp bracket process hS Tm i of S Tm ,
which exists under Qm since S Tm ∈ M2 (Qm ). The reason for not working with [S Tm ] (which
always exists, even under P ) is that in the construction of h̃i in step a) of the proof, we have
to ensure that h̃i is predictable. For the one-dimensional case d = 1, one could define h̃i
directly from h; but for d > 1, the bracket process of S Tm also comes up via the matrix σ m ,
and so we have to work with hS Tm i because [S Tm ], as well as its corresponding matrix, is
only optional in general. A similar issue comes up in the construction of the version ϑ(m) at
the end of the proof of Lemma 3.3.
¦
Proposition 3.10. Take the version of ϑ(m) from Lemma 3.3. Viewed as a process on Ω̄,
each coordinate t 7→ ϑ(m),k (t, ·) is then a (real-valued) quasimartingale with respect to (Pt )
37
¤
¡
¤¢
£
£
and Qm ⊗ (S Tm )k . Thus it has a version which is Qm ⊗ (S Tm )k -almost everywhere
right-continuous in t.
Proof. To simplify notation, we once again drop all indices m and work with Tm = T ,
writing also for brevity S instead of S Tm . Fix an index k ∈ {1, . . . , d} throughout.
1) To show that ϑk is a quasimartingale, we need to prove that it is adapted, integrable,
and has the quasimartingale property (see below). Adaptedness is clear since Pt ≡ P and
each ϑk (t, ·) is by definition a predictable process on Ω, and integrability will be argued
presently. The main point is to show that
·X
¸
¯
¯
k
¯EQ⊗[S k ] [∆i+1 ϑ | Pt ]¯ < ∞,
sup EQ⊗[S k ]
i
(3.31)
Π
ti ∈Π
where ∆i+1 ϑk = ϑk (ti+1 , ·) − ϑk (ti , ·) as before and the supremum is taken over all partitions Π of [0, T ]. But Pti ≡ P; therefore the (inner) conditional expectation vanishes,
and only the (outer) expectation remains. Moreover, for any given partition Π, setting
P
sign(∆i+1 ϑk )I(ti ,ti+1 ] gives a very simple real-valued predictable process on Ω̄ which
f :=
ti ∈Π
is bounded by 1. Because f . ϑkT =
P
sign(∆i+1 ϑk )∆i+1 ϑk =
ti ∈Π
ti ∈Π
(3.30) from (the proof of) Proposition 3.7 gives
·X
¸
¯
¯
°
°
k¯
¯
∆i+1 ϑ
= °f . ϑkT °
EQ⊗[S k ]
ti ∈Π
≤ kf k∞
³
P
|∆i+1 ϑk | ≥ 0, the estimate
L1 (Q⊗[S k ])
°
°1
2
kSTk kL2 (Q) °B̃T °∞
+ kJ
(1)
°
° 21 ´
°
kL2 (Q) |Ã|T °∞ .
But kf kL∞ = 1, and so the right-hand side above does not depend on Π, which proves
(3.31) and etablishes the quasimartingale property of ϑk in t. Finally, integrability follows
¢
¡
in the same way, by choosing f := sign ϑk (t, ·) I(0,t] to obtain that f . ϑkT = |ϑk (t, ·)| is in
L1 (Q ⊗ [S k ]).
2) Exactly as in the proof of Rao’s theorem (see Dellacherie/Meyer (1982), Theorem
VI.40), the quasimartingale ϑk can be decomposed as the difference of two supermartingales
ψ ± . Both of these have (Q⊗[S k ])-a.e. right limits (in the argument t) along the rationals, and
because Pt ≡ P is obviously right-continuous, the processes ϕ± obtained as the right limits
along the rationals are again supermartingales (see Dellacherie/Meyer (1982), Theorem VI.2)
and of course right-continuous in t by construction. But now the argument goes as in Section
2 for Proposition 2.13: The process ϑk = ψ + − ψ − admits right limits along the rationals
since ψ ± do, and the process obtained as those limits equals ϕ+ − ϕ− and is right-continuous.
But we also know from Lemma 3.3 that ϑk is right-continuous in measure (with respect to
µm , hence with respect to Q ⊗ [S k ] ¿ µm as well), and so ϕ+ − ϕ− is also a version of ϑk by
(the proof of) Lemma 2.2. This completes the argument.
q.e.d.
38
Putting everything together now readily yields the main result of this section.
Theorem 3.11. As in Theorem 2.14, suppose that the IRd -valued semimartingale S satisfies
(2.9), and U satisfies (2.11)–(2.13). Viewed as a process on Ω̄ = Ω × (0, ∞), each coordinate
¤¢
£
¡
t 7→ ϑ(m),k (t, ·) has a version which is Qm ⊗ (S Tm )k -almost everywhere right-continuous
and of finite variation in t.
Proof. Take the right-continuous version from Proposition 3.10. This does not change the
integrals, with respect to the vector-valued ϑ(m) , of very simple matrix-valued predictable
processes on Ω̄, so that the conclusions of Proposition 3.7 hold for this version as well. Thus
the Bichteler–Dellacherie characterisation of semimartingales (see Dellacherie/Meyer (1982),
Theorem VIII.80) implies for each k that this version t 7→ ϑ(m),k (t, ·) is on Ω̄ a semimartingale;
note that this uses right-continuity of ϑ(m),k in t, which is why we had to establish Proposition
3.10 beforehand. But the filtration Pt ≡ P on Ω̄ is constant; hence ϑ(m),k has no nontrivial
martingale part and is therefore of finite variation (in the t-argument).
q.e.d.
Remark 3.12. Because ϑ(m) (t, u) = ϑ̂(t ∧ Tm , u ∧ Tm ) is not simply “the process ϑ̂ stopped
at Tm ”, it is not clear whether we can easily (or at all) formulate an analogous result for the
process ϑ̂ itself. The problem is that while Tm acts on the u-argument in the usual “stopping
way”, its effect on the t-argument is less clear — recall that this first argument is the time
¦
horizon of a utility maximisation problem.
If we forget the financial origin of the problem studied in this paper, we can view Theorems 2.14 and 3.11 as providing an answer to the following general question. Suppose
¡
¢
Y = (Yt )t≥0 is a semimartingale and ψ(t, u) t≥0, 0≤u≤t is a two-parameter process such that
¡
¢
for each fixed t, the process ψ(t, u) 0≤u≤t is predictable and Y -integrable on [0, t]. For each
t ≥ 0, the random variable
Zt = ψ(t, ·) . Yt =
Zt
ψ(t, u) dYu
0
is hence well defined. In that abstract setting, when is Z = (Zt )t≥0 a semimartingale? More
precisely, there are two questions:
1) Which conditions on ψ and Y imply that Z is a semimartingale?
2) Conversely, if Z is a semimartingale, what does this imply about ψ?
Even more generally, one can ask about the properties of a process Z that arises in such a
way from a pair (ψ, Y ). At this level of generality, the question is too vague. If Y is Brownian
motion and ψ is a nonrandom function, then Z is a Volterra-type integral process and there
are many Gaussian-type results; see for instance Chapter VI in Hida/Hitsuda (1993). If in
39
R
addition ψ(t, u) = f (t − u) for some deterministic function f , then Z = f (· − u) dWu is
a convolution-type integral or moving average of Brownian motion W . In that specific case,
questions 1) and 2) have very precise answers; we refer to the recent work of Basse/Pedersen
(2009) which generalises these classical results from Brownian motion to the case where Y is
a Lévy process, and which also contains a very good overview of earlier and recent literature.
But for a general semimartingale Y and a general (stochastic) ψ, results seem to be much
more sparse. For the easier question 1), there is an affirmative result in Protter (1985) under
a rather restrictive assumption on ψ; see the end of Section 4 for a more detailed discussion.
Concerning question 2) at the general semimartingale level, we are not aware of any results
in the literature so far.
If one looks more carefully at the arguments that lead to Theorem 3.11, one can see that
our approach actually provides a partial answer to question 2) in the following way. All we
really need for our arguments is that locally, after an equivalent change of measure, Y is a
square-integrable martingale, all stochastic integral processes ψ(t, ·) . Y can be dominated in
L2 , and the semimartingale Z is in S 2 . We could formulate this as an abstract theorem and
it seems very likely that these ideas can be further extended, but we leave this issue for future
work.
4. A new stochastic Fubini theorem
In this section, we establish a new type of weak stochastic Fubini theorem. This is needed in
e in more explicit form. However, the
order to obtain in the next section the structure of X
result in this section is also of independent interest in itself.
The key idea for the argument in the next section is to view the process ϑ̂ as a process
on Ω, indexed by u and taking values in the space of signed measures on [0, ∞); this is
justified by Theorem 3.11 since (a good version of) each ϑ(m) (·, u) is of finite variation with
respect to t (at least after suitable “stopping” with Tm ). Moreover, we integrate in the next
section with respect to S Tm which is a Qm -square-integrable Qm -martingale. The abstract
formulation we need thus involves integrals of the form ϕ . M with a martingale M and a
measure-valued process ϕ. Our construction of ϕ . M is strongly inspired by Björk/Di Masi/
Kabanov/Runggaldier (1997), but there are important differences that we explain below.
(∞)
the space of all signed (possibly
So fix a time horizon T ∈ (0, ∞), denote by IMT
infinite) measures m on [0, T ] with m({0}) = 0 and by CT = C([0, T ]; IR) the space of all
continuous functions f : [0, T ] → IR with the sup-norm kf k∞ . Denote the integral of f with
respect to m by
m(f ) := f . mT =
ZT
f (t) m(dt)
0
40
(∞)
for f ∈ CT , m ∈ IMT
(∞)
and write kmkv := sup {m(f ) | f ∈ CT with kf k∞ ≤ 1} for the variation norm of m ∈ IMT
.
(∞)
IMT
the σ-field MT generated by the weak∗ topology, i.e. by all mappings
©
ª
(∞) ¯
m 7→ m(f ) for f ∈ CT . Finally, we set IMT := m ∈ IMT ¯ kmkv < ∞ . For d ∈ IN and
m = (mi )i=1,...,d ∈ IMTd , we use the notation kmkv := (kmi kv )i=1,...,d ∈ IRd .
We take on
A process ϕ = (ϕu )0≤u≤T defined on (Ω, F, P ) and with values in IMTd is called weakly
¢
¡
predictable if the IRd -valued process ϕ(f ) = ϕu (f ) 0≤u≤T is predictable for each test function
f ∈ CT . In view of the definition of MT , this is the same as saying that as a mapping from
Ω̄ = Ω × (0, ∞) (or, more precisely, Ω̄T = Ω × (0, T ]) to IMTd , ϕ is P-MdT -measurable. We
remark for later use that the IRd -valued process kϕkv = (kϕu kv )0≤u≤T is then predictable.
Indeed, because CT is separable for the sup-norm, we can take a countable dense subset
(fk )k∈IN of CT and obtain sup G(f ) = sup G(fk ) for any continuous function G : CT → IR.
i
f ∈CT
k∈IN
Taking G(f ) = m (f ) for i = 1, . . . , d thus yields for each (ω, u) ∈ Ω̄ that
kϕi kv (ω, u) = kϕiu kv (ω) = sup
f ∈CT
¢
¢
1 ¡ i
1 ¡ i
ϕu (f ) (ω) = sup
ϕu (fk ) (ω).
kf k∞
k∈IN kfk k∞
Since each ϕ(fk ) is predictable, so is then kϕkv as a countable supremum.
Now let M = (Mu )0≤u≤T be an IRd -valued martingale in M20 (P ) and denote by E b the
family of all IMTd -valued processes ϕ of the form
ϕ=
L−1
X
`=0
m` ID` ×(u` ,u`+1 ]
with L ∈ IN , 0 ≤ u0 < u1 < · · · < uL ≤ T , m` ∈ IMTd and D` ∈ Fu` . For each f ∈ CT ,
L−1
P
m` (f )ID` ×(u` ,u`+1 ] is then a very simple IRd -valued predictable process (on Ω),
ϕ(f ) =
`=0
which justifies calling ϕ ∈ E b a very simple IMTd -valued weakly predictable process (on Ω̄).
Similarly, we call an IMT -valued process N = (Nu )0≤u≤T a weak martingale in M20 (P ) if
¡
¢
the real-valued process N (f ) = Nu (f ) 0≤u≤T is in M20 (P ) for each f ∈ CT . To define an
IMT -valued stochastic integral ϕ . M for general IMTd -valued ϕ, we start with ϕ ∈ E b and then
extend to a larger class of integrands. Note that while both ϕ and M are d-dimensional (like
b
ϑ̂ and S, respectively), the resulting integral ϕ . M will be one-dimensional (like X).
Lemma 4.1. Fix an IRd -valued martingale M in M20 (P ). For each ϕ ∈ E b , there exists a
process ϕ . M = (ϕ . Mu )0≤u≤T with values in IMT which is a weak martingale in M20 (P ) and
which has the weak Fubini property
(4.1)
(ϕ . Mu )(f ) = ϕ(f ) . Mu ,
0 ≤ u ≤ T , for each f ∈ CT .
41
Written out in terms of integrals, (4.1) takes the form
ZT
f (t)
0
µ Zu
¶
ϕs dMs (dt) =
0
Zu µ ZT
0
¶
0 ≤ u ≤ T,
f (t) ϕs (dt) dMs ,
0
which explains the terminology. In particular, we have the isometry property
(4.2)
k(ϕ . M )(f )kM2 (P ) = k(ϕ . MT )(f )kL2 (P ) = kϕ(f ) . MT kL2 (P ) = kϕ(f )kL2 (P,M )
for f ∈ E b .
Proof. Writing ϕ =
L−1
P
`=0
m` ID` ×(u` ,u`+1 ] , it is clear that we shall set
ϕ . Mu :=
L−1
X
`=0
ID` (Mu`+1 ∧u − Mu` ∧u )tr m` ,
0≤u≤T
which clearly has values in IMT . Then we get for each f ∈ CT that
(4.3)
(ϕ . Mu )(f ) =
L−1
X
`=0
ID` m` (f )tr (Mu`+1 ∧u − Mu` ∧u ) =
Zu
hs dMs
0
with the process
(4.4)
hs (ω) :=
L−1
X
m` (f )ID` (ω)I(u` ,u`+1 ] (s)
`=0
=
T
L−1
XZ
`=0 0
=
µ ZT
ID` ×(u` ,u`+1 ] (ω, s)f (t) m` (dt)
¶
f (t) ϕs (dt) (ω)
0
= ϕs (f )(ω).
Moreover, the predictable process h = ϕ(f ) is bounded uniformly in ω, u, by
(4.5)
¯
ª
©
C := max kmi` kv kf k∞ ¯ i = 1, . . . , d, ` = 0, . . . , L − 1 ,
and hence it is in L2 (P, M ). Now (4.3) immediately shows that (ϕ . M )(f ) is a martingale
and in M20 (P ), and combining (4.3) with (4.4) also directly gives (4.1). Finally, (4.2) follows
from (4.1).
q.e.d.
42
Remark 4.2. The above construction and setup as well as the subsequent extension to
general ϕ are strongly inspired by the work of Björk/Di Masi/Kabanov/Runggaldier (1997).
However, there is an important difference. In Björk/Di Masi/Kabanov/Runggaldier (1997),
the goal is to develop stochastic integration with respect to a process (M̄ , say) with values in
CT ; the natural integrands ϕ are then IMT -valued and the resulting integral ϕ . M̄ has values
in IR. In contrast, our integrator M is IRd - or real-valued, and so using the same IMTd - or
IMT -valued integrands ϕ leads to an integral process ϕ . M which is again measure-valued . ¦
To extend both ϕ . M and the weak Fubini property (4.1) from ϕ ∈ E b to more general
integrands, we need appropriate seminorms on ϕ and on weak martingales N like ϕ . M . Recall
that (fk )k∈IN is a countable dense subset of CT .
Definition 4.3. For an IMT -valued weak martingale N = (Nu )0≤u≤T in M20 (P ), we define
¶ 21
µX
∞
2
γk kN (fk )kM2 (P ) ,
rγ (N ) :=
k=1
where the sequence γ = (γk )k∈IN of weights in (0, ∞) will be chosen later. We then denote
by Nγ2 the space of all IMT -valued weak martingales N in M20 (P ) with rγ (N ) < ∞.
Definition 4.4. For an IMTd -valued weakly predictable process ϕ = (ϕu )0≤u≤T , we define
¶ 12
µX
∞
2
γk kϕ(fk )kL2 (P,M ) ,
qγ (ϕ) :=
k=1
where the sequence γ = (γk )k∈IN of weights in (0, ∞) will be chosen later. We then denote
by L2BDKR the space of all IMTd -valued weakly predictable processes ϕ with qγ (ϕ) < ∞.
Because M is IRd -valued and in M20 (P ), we can represent its (matrix-valued) bracket
R
process similarly as in Section 3 in the form hM i , M j i = σuij dBu for i, j = 1, . . . , d, with an
increasing predictable integrable RCLL process B null at 0 and a predictable process σ taking
values in the symmetric nonnegative definite d × d-matrices. (Of course B and σ here are
not the same as in Section 3, but this should not lead to any confusion.) For any IRd -valued
predictable process ψ, we then have
" ZT
#
(4.6)
kψk2L2 (P,M ) = E[hψ . M iT ] = E
ψutr σu ψu dBu = kψ tr σψkL1 (P ⊗B) .
Moreover, |σ ij | ≤
√
0
σ ii σ jj by the Kunita–Watanabe inequality, and combining this with the
Cauchy–Schwarz inequality yields
tr
|ψ σψ| ≤
d
X
i,j=1
i ij
j
|ψ σ ψ | ≤
43
d kψk2∞
d
X
i=1
σ ii
for any bounded IRd -valued predictable process ψ. Integrating with respect to B and taking
expectations gives
(4.7)
tr
kψ σψkL1 (P ⊗B) ≤
d kψk2∞
d
X
i=1
°
°2
E[hM i iT ] = d kψk2∞ °|MT |°L2 (P ) < ∞
because M is in M20 (P ). This will be useful later on.
Lemma 4.5. Suppose γk , k ∈ IN , are in (0, ∞) and satisfy
∞
P
k=1
γk < ∞ and
∞
P
k=1
γk kfk k2∞ ≤ 1.
(Clearly, such sequences γ exist.) Then:
1) rγ and qγ are seminorms.
2) For each b > 0 and with 1 = (1, . . . , 1) ∈ IRd , the functional qγ is continuous with
respect to the weak∗ topology on the subspace of L2BDKR consisting of those ϕ taking values
¯
©
ª
in Ubd := m ∈ IMTd ¯ kmkv ≤ b1 : If (ϕn )n∈IN is a sequence in L2BDKR with values in Ubd and
converging (P ⊗ B)-almost everywhere to some Ubd -valued ϕ for the weak∗ topology, then
lim qγ (ϕn ) = qγ (ϕ).
n→∞
3) For ϕ ∈ E b , we have the isometry
rγ (ϕ . M ) = qγ (ϕ).
(4.8)
Proof. 1) It is clear that rγ and qγ are both positively homogeneous. For the triangle
inequality, we start by noting that any m ∈ IMT (or IMTd ) can be viewed as a continuous
(IRd -valued) linear functional on CT via f 7→ m(f ) and is hence determined by the sequence
¢
¡
m(fk ) k∈IN of its values on the dense subset (fk )k∈IN of CT . Hence N ∈ Nγ2 can be identified
¢
¡
with the sequence of random variables NT (fk ) k∈IN , since each N (fk ) is a martingale, and
rγ2 (N )
=
∞
X
k=1
γk kNT (fk )k2L2 (P )
=
∞
X
γk x2k
k=1
with xk := kNT (fk )kL2 (P ) , and analogously for N̄ with x̄. This shows that rγ (N ) is simply
the `2 -norm of the sequence x for the measure having weights γk , k ∈ IN , on IN . But
yk := kNT (fk ) + N̄T (fk )kL2 (P ) ≤ xk + x̄k
by the usual triangle inequality in L2 (P ), and this gives
rγ (N + N̄ ) = kyk`2 (γ)
µX
¶ 12
∞
2
≤
γk (xk + x̄k )
k=1
= kx + x̄k`2 (γ)
≤ kxk`2 (γ) + kx̄k`2 (γ)
= rγ (N ) + rγ (N̄ ).
44
The argument for qγ is completely analogous.
2) Take a sequence (ϕn )n∈IN of weakly predictable processes with kϕn kv ≤ b1 for all n
and suppose that (ϕn ) converges to ϕ for the weak∗ topology (P ⊗ B)-a.e. This implies that
lim ϕnu (fk ) = ϕu (fk ) for all k ∈ IN (P ⊗ B)-a.e. and hence also (P ⊗ B)-a.e.
n→∞
lim ϕnu (fk )tr σu ϕnu (fk ) = ϕu (fk )tr σu ϕu (fk )
n→∞
for all k ∈ IN .
√
¯
¯
But |ϕnu (fk )| ≤ ¯kϕnu kv ¯kfk k∞ ≤ d bkfk k∞ (P ⊗ B)-a.e. and so we can apply dominated
¢
¡
convergence to obtain that ϕnu (fk )tr σu ϕnu (fk ) n∈IN tends to ϕu (fk )tr σu ϕu (fk ) in L1 (P ⊗ B)
for all k ∈ IN , which means by (4.6) that lim ϕn (fk ) = ϕ(fk ) in L2 (P, M ) for all k ∈ IN .
n→∞
Moreover, Fatou’s lemma yields for all k ∈ IN that
kϕ(fk )k2L2 (P,M ) = kϕ(fk )tr σϕ(fk )kL1 (P ⊗B)
≤ lim inf kϕn (fk )tr σϕn (fk )kL1 (P ⊗B)
n→∞
≤ sup kϕn (fk )tr σϕn (fk )kL1 (P ⊗B)
n∈IN
°
°2
≤ d2 b2 kfk k2∞ °|MT |°L2 (P )
thanks to (4.7) since each ϕn has values in Ubd . So if we first take K large enough to obtain
∞
°
°2
P
γk kfk k2∞ ≤ ε2 and then n large enough to get
4d2 b2 °|MT |°L2 (P )
k=K+1
γk kϕn (fk ) − ϕ(fk )k2L2 (P,M ) ≤ ε2 /K
for k = 1, . . . , K,
we obtain from the definition of qγ and the above estimates that
qγ2 (ϕn
− ϕ) =
∞
X
k=1
γk kϕn (fk ) − ϕ(fk )k2L2 (P,M ) ≤ 2ε2 .
This proves the assertion.
3) The isometry (4.2) in Lemma 4.1 yields kϕ(fk )k2L2 (P,M ) = k(ϕ . M )(fk )k2M2 (P ) for
ϕ ∈ E b and all k ∈ IN . Hence (4.8) follows directly from the definitions of rγ and qγ . q.e.d.
Lemma 4.6. If we choose the γk as in Lemma 4.5, then E b is dense in L2BDKR with respect
to the seminorm qγ .
Proof. This is very similar to Lemma 2.3 of Björk/Di Masi/Kabanov/Runggaldier (1997),
but for completeness we give the proof in detail. First of all, we have E b ⊆ L2BDKR ; indeed,
L−1
P
m` ID` ×(u` ,u`+1 ] is in E b , then |ϕ(f )| ≤ Ckf k∞ with a constant C like in (4.5)
if ϕ =
`=0
45
only depending on ϕ, and so we obtain qγ (ϕ) < ∞ by using (4.6), (4.7), the definition of
∞
P
γk kfk k2∞ < ∞. To prove denseness, note that by Banach–Alaoglu, each ball
qγ and
k=1
¯
©
ª
Ubd := m ∈ IMTd ¯ kmkv ≤ b1 is compact in the weak∗ topology on IMTd and metrisable.
Thus any P-MdT -measurable mapping ϕ : Ω̄ → Ubd can be approximated in the sense of weak∗
J
P
convergence by P-measurable step functions of the form ϕn =
mj IAj with mj ∈ IMTd and
¡
Aj ∈ P. Hence ϕn (f )
¢
j=1
n∈IN
converges to ϕ(f ) (P ⊗ B)-a.e. for every f ∈ CT , and part 2) of
Lemma 4.5 then yields that (ϕn ) tends to ϕ in the seminorm qγ since all the ϕn are chosen
to have values in Ubd like ϕ.
Now the predictable σ-field P is generated by the sets of the form D` × (u` , u`+1 ] with
L−1
P
D` ∈ Fu` , so that the real-valued predictable processes of the form g =
c` ID` ×(u` ,u`+1 ] are
`=0
2
dense in L (P, P ⊗ B). By passing to a subsequence, we can find for any A ∈ P a sequence
(g n )n∈IN of the above form such that g n → IA (P ⊗ B)-a.e. For m ∈ Ubd and f ∈ CT , we then
get first m(f )g n → m(f )IA in IRd (P ⊗ B)-a.e. and then also in L2 (P, M ) by using (4.6),
(4.7) and dominated convergence. Because the same applies to finite linear combinations, we
see that E b is dense, for the L2 (P ⊗ B)-norm, in the set of all P-measurable step functions
ϕn as used above. But in view of the definition of qγ , this is enough to conclude that E b is
dense, with respect to qγ , in the set of all weakly predictable Ubd -valued processes, and hence
also in L2BDKR .
q.e.d.
Lemma 4.7. If we choose the γk as in Lemma 4.5, then Nγ2 is complete with respect to the
seminorm rγ .
Proof. This is fairly straightforward. If (N n )n∈IN is Cauchy in Nγ2 with respect to rγ , the
¢
¡
definition of rγ implies that N n (fk ) n∈IN is Cauchy in M2 (P ) for each k ∈ IN and thus
convergent to a limit process in M20 (P ) that we call N ∞,k . Since each IMT -valued weak
martingale L in M2 (P ) is determined by the values of LT (fk ), k ∈ IN , we can define such a
weak martingale N ∞ by setting NT∞ (fk ) := NT∞,k . For each finite K, we then get
K
X
k=1
n
γk kN (fk ) − N
∞
(fk )k2M2 (P )
= lim
m→∞
K
X
k=1
≤ lim sup
m→∞
γk kN n (fk ) − N m (fk )k2M2 (P )
∞
X
k=1
γk kN n (fk ) − N m (fk )k2M2 (P ) ≤ ε2
for large n by the Cauchy property in Nγ2 , and this shows by letting K → ∞ that we have
rγ (N n − N ∞ ) ≤ ε for n large enough. So N ∞ lies in Nγ2 and (N n ) converges to N ∞ with
respect to rγ . This proves the assertion.
q.e.d.
46
Putting everything together, we can now establish the main result of this section.
Theorem 4.8. Fix an IRd -valued martingale M in M20 (P ), choose a countable dense subset
∞
P
(fk )k∈IN of CT and let γ = (γk )k∈IN be a sequence in (0, ∞) satisfying
γk < ∞ and
k=1
∞
P
k=1
γk kfk k2∞ ≤ 1. For every ϕ ∈ L2BDKR , the process ϕ . M is then well defined, in Nγ2 , and
we have the weak Fubini property
(4.9)
(ϕ . Mu )(f ) = ϕ(f ) . Mu ,
0 ≤ u ≤ T , for each f ∈ CT .
In particular, ϕ . M is an IMT -valued weak martingale in M20 (P ).
Proof. The mapping ϕ 7→ ϕ . M from E b to Nγ2 is by construction linear, and due to the
isometry in part 3) of Lemma 4.5 continuous with respect to qγ and rγ . Moreover, E b is dense
in L2BDKR with respect to qγ by Lemma 4.6, and Nγ2 is complete with respect to rγ by Lemma
4.7. Therefore we can extend the above mapping in a unique way to a linear mapping from
L2BDKR to Nγ2 , and the isometry (4.8) still holds for that extension by continuity so that
(4.10)
rγ (ϕ . M ) = qγ (ϕ)
for all ϕ ∈ L2BDKR .
To show that the weak Fubini property (4.1) extends from ϕ ∈ E b to ϕ ∈ L2BDKR as well,
¡
we first note that for each f ∈ CT , both sides of (4.1) and (4.9) are martingales even in
¢
M20 (P ) so that it is enough to argue for u = T . By the separability of CT , it is also enough
to take only fk , k ∈ IN , instead of all f ∈ CT , so that we need to show for all k ∈ IN and for
ϕ ∈ L2BDKR that
(4.11)
(ϕ . MT )(fk ) = ϕ(fk ) . MT .
So take (ϕn )n∈IN in E b with qγ (ϕn − ϕ) → 0 as n → ∞. Then ϕn (fk ) → ϕ(fk ) in L2 (P, M )
for each k by the definition of qγ , and so ϕn (fk ) . MT → ϕ(fk ) . MT in L2 (P ) for each k
by the usual isometry property of stochastic integrals. But due to (4.10) (or from the very
construction), we also have rγ (ϕn . M − ϕ . M ) → 0 as n → ∞, which implies by the definition
of rγ that (ϕn . MT )(fk ) → (ϕ . MT )(fk ) in L2 (P ) for each k. Because (4.11) holds for each ϕn
by Lemma 4.1, we thus conclude that it holds for ϕ as well, and this ends the proof. q.e.d.
For our subsequent application, we need to extend the weak Fubini property (4.9) from
f ∈ CT to f being an indicator function of an interval. This can be done at the cost of slightly
stronger assumptions on ϕ; but these will be satisfied in our application, thanks to part 2)
of Proposition 3.7.
47
Theorem 4.9. Under the same assumptions as in Theorem 4.8, let ϕ be an IMTd -valued
weakly predictable process satisfying
kϕ(f )kL2 (P,M ) ≤ Ckf k∞
(4.12)
for all f ∈ CT
with a constant C not depending on f . Then ϕ is in L2BDKR , and for any bounded measurable
function f on [0, T ], (ϕ . M )(f ) is a martingale in M20 (P ) and the weak Fubini property (4.9)
holds. In particular, for f = I[0,t] , we get
Zt µ Zu
0
¶
ϕs dMs (dr) =
0
Zu µ Zt
0
¶
0 ≤ t ≤ T, 0 ≤ u ≤ T.
ϕs (dr) dMs ,
0
Proof. Using the definition of qγ and (4.12) gives
qγ2 (ϕ)
=
∞
X
k=1
γk kϕ(fk )k2L2 (P,M )
≤C
2
∞
X
k=1
γk kfk k2∞ < ∞
by the choice of γ so that ϕ is indeed in L2BDKR . Moreover, ϕu (f ) =
RT
f (t) ϕu (dt) is well
0
defined for (P ⊗B)-almost all (ω, u) for every bounded measurable function f on [0, T ] because
¯
¯
¯kϕu kv ¯ < ∞ (P ⊗ B)-a.e. since ϕ has values in IM d .
T
Now take any bounded measurable function f on [0, T ] and choose a sequence (f n )n∈IN
in CT with f n → f pointwise and kf n k∞ ≤ kf k∞ . Then ϕ(f n ) → ϕ(f ) in IRd by dominated
convergence, and so (ϕ(f n )tr σϕ(f n ) → ϕ(f )tr σϕ(f ) (P ⊗ B)-a.e. As a consequence, (4.12)
extends from f ∈ CT to any bounded real-valued measurable function f on [0, T ]; indeed,
combining Fatou’s lemma with (4.6) and (4.12) for f n yields
kϕ(f )k2L2 (P,M ) = kϕ(f )tr σϕ(f )kL1 (P ⊗B)
≤ lim inf kϕ(f n )k2L2 (P,M )
n→∞
≤ lim inf C 2 kf n k2∞
n→∞
≤ C 2 kf k2∞ .
Applying this to f n − f yields ϕ(f n ) → ϕ(f ) in L2 (P, M ) since kf n − f k∞ → 0, and so the
usual isometry property of stochastic integrals gives ϕ(f n ) . M → ϕ(f ) . M in M2 (P ) and also
ϕ(f n ) . MT −→ ϕ(f ) . MT
in L2 (P ).
In particular, ϕ(f ) . M is again a martingale in M20 (P ). But on the other hand, ϕ . MT is by
construction in IMT . Hence f n → f implies as above by dominated convergence that
(ϕ . MT )(f n ) −→ (ϕ . MT )(f )
48
P -a.s.,
and we have (ϕ . MT )(f n ) = ϕ(f n ) . MT for each f n by (4.9) since f n ∈ CT . Hence uniqueness
of the limit gives (ϕ . MT )(f ) = ϕ(f ) . MT . Replacing f by f I[0,u] gives in the same way that
(ϕ . Mu )(f ) = ϕ(f ) . Mu , and so (ϕ . M )(f ) is a martingale in M20 (P ) because ϕ(f ) . M is. This
completes the proof.
q.e.d.
At this point, it seems appropriate to comment on how our stochastic Fubini theorem
and in particular Theorem 4.9 relate to the existing literature. The typical formulation is
that one has a semimartingale Y , a parametric family ψ x of processes with x from some
space X , and a measure η on X . For each x, the process ψ x lies in L(Y ), and so does (by
R
assumption) ψ̄ η := ψ x η(dx). The typical stochastic Fubini theorem then has the form
Z ³Z
´
x
ψ η(dx) dY =
Z
η
ψ̄ dY =
Z ³Z
x
´
ψ dY η(dx);
see for instance Théorème 5.44 of Jacod (1979), Lebedev (1995) [which corrects an error in
Jacod (1979)], Theorem IV.65 of Protter (2005), or Bichteler/Lin (1995) [which extends the
result in Protter (2005) from an L2 - to an L1 -setting]. However, all these results share the
restrictive assumption that one has a fixed measure η which only depends on the parameter
x and not on the randomness ω. For comparison, our setting in Section 3 has X =
b [0, T ] and
b ϑ(t∧Tm , ω, u∧Tm ) with x =
b t. However, while each ϑ(·∧Tm , ω, u∧Tm ) is by Theorem
ϕxu (ω) =
3.11 of finite variation and can hence be used as an integrator, there need not exist any
measure on [0, T ] with respect to which all the ϑ(· ∧ Tm , ω, u ∧ Tm ) are absolutely continuous.
Hence already our basic framework is considerably more complicated, and establishing a
stochastic Fubini theorem therefore needs quite different, new techniques.
The paper that is probably closest to our basic question is Protter (1985). The problem
studied there is to establish the semimartingale property and decomposition for a stochastic
Rt
Volterra integral process of the form t 7→ Zt := ψ(t, u) dYu for a semimartingale Y . This
0
¡
looks indeed very similar to (1.5), and the idea that we borrowed in (1.5) from Protter
¢
(1985) is to decompose Zt as
Zt
0
ψ(t, u) dYu =
Zt
ψ(u, u) dYu +
0
Zt
0
¡
¢
ψ(t, u) − ψ(u, u) dYu
and to use a stochastic Fubini theorem. But the crucial assumption in Protter (1985) is that
t 7→ ψ(t, u) is in C 1 (with a locally Lipschitz derivative)
so that one has
ψ(t, u) − ψ(u, u) =
49
Zt
u
∂ψ
(r, u) dr.
∂t
This means that with respect to the first parameter t, all the ψ(t, u) are absolutely continuous
with respect to one fixed (namely Lebesgue) measure, and so things can be reduced to a
“standard” stochastic Fubini theorem. This is not possible in our situation, so that our
generalisation is really needed.
Remark 4.10. While our stochastic Fubini theorem generalises earlier work with regard
to the allowed integrands, we should in fairness also mention that we are in contrast a bit
more restrictive about the integrator, M , which is a martingale. We are confident that we
can extend our approach to a semimartingale; but the martingale case is sufficient for our
needs here, and also (as far as we can see) already contains almost all the important new
ideas. Therefore (and also for reasons of space) we have decided to postpone the more general
¦
semimartingale case to future work.
5. The structure of the process of optimal wealths
b of optimal final wealths has a version X
e which
We have seen in Section 2 that the process X
b the process X
e has the form
is a semimartingale. From the definition of X,
et = x +
X
Zt
ϑ̂(t, u) dSu ,
0
t≥0
which is a Volterra-type of stochastic integral. With the stochastic Fubini theorem from
Section 4, we now have everything we need to find the structural decomposition of the semie
martingale X.
Throughout this section, we again assume that the IRd -valued semimartingale
S satisfies (2.9), and U satisfies (2.11)–(2.13), so that we can use all the results from
Sections 2 and 3. We start with an arbitrary but fixed T ∈ (0, ∞) and construct the stopping
times (Tm )m∈IN and probability measures (Qm )m∈IN via Lemma 3.1. Then we write for fixed
m and 0 ≤ t ≤ T
(5.1)
etTm = X
btTm = x +
X
=x+
t∧T
Z m
0
Zt
0
+
Zt
0
ϑ̂(t ∧ Tm , u) dSu
¡
¢
ϑ̂(t ∧ Tm , u ∧ Tm ) − ϑ̂(u ∧ Tm , u ∧ Tm ) dSuTm
ϑ̂(u ∧ Tm , u ∧ Tm ) dSuTm .
50
We recall that ϑ̂(t ∧ Tm , u ∧ Tm ) = ϑ(m) (t, u) and take for ϑ(m) the version from Proposition
3.10. Then Corollary 3.6 tells us that
ϑ̂(u ∧ Tm , u ∧ Tm ) = ϑ(m) (u, u) = ϑ̄(m)
u ,
0≤u≤T
is predictable and S Tm -integrable so that the last term in (5.1) is well defined and equals
Zt
(5.2)
ϑ̄u(m) dSuTm =
0
Zt
0
ϑ̂(u ∧ Tm , u ∧ Tm )I{u≤Tm } dSu =
t∧T
Z m
ϑ̂(u, u) dSu .
0
For the first term, we recall from Theorem 3.11 that each coordinate of ϑ(m) is, as a
function of its first argument t, right-continuous and of finite variation. So the “random
signed IRd -valued distribution functions”
b 7→ ϕ̄u(m) (b) := ϑ(m) (b, u) − ϑ(m) (u, u),
(5.3)
0≤u≤T
on [0, ∞) can be associated with an IMTd -valued process ϕ(m) , which is clearly weakly predict¢
(m) ¡
able since ϕu (0, t] = ϑ(m) (t, u) − ϑ(m) (0, u) = ϑ(m) (t, u) is predictable as a process in u.
(m)
Moreover, ϕu ({0}) = ϑ(m) (0, u) = 0 since any ϑ(m) (t, u) vanishes at u = 0 and we can only
take u ≤ t (= 0 here); and part 2) of Proposition 3.7 tells us that
kϕ(m) (f )kL2 (Qm ,S Tm )
° ZT
°
°
°
°
°
(m)
= ° f (t) ϑ (dt, ·)°
°
°
0
L2 (Qm ,S Tm )
°
(m) °
= °f . ϑT °L2 (Qm ,S Tm ) ≤ Cm kf k∞
for each bounded measurable real-valued function f on [0, T ]. More precisely, we get this
inequality from Proposition 3.7 for piecewise constant measurable f ; but as pointed out in
Remark 3.9, it then extends directly to all bounded measurable f on [0, T ] because we know
now that ϑ(m) is of finite variation in t. Therefore the condition (4.12) in Theorem 4.9 is
also satisfied, and this allows us to handle the more complicated first term in (5.1) via our
stochastic Fubini theorem.
Lemma 5.1. For each m ∈ IN , the real-valued process
(m)
Bt
:=
Zt
0
¡
¢
ϑ(m) (t, u) − ϑ(m) (u, u) dSuTm ,
0 ≤ t ≤ T,
is RCLL, of finite variation and predictable.
Proof. Because ϑ(m) (t, u) vanishes for u > t, we can write
(m)
Bt
=
ZT
0
¡
ϑ
(m)
(t, u) − ϑ
(m)
(u, u)
¢
dSuTm
=
ZT
0
51
¡
¢
ϕu(m) (I(0,t] ) dSuTm = ϕ(m) (I(0,t] ) . STTm .
(m)
But the definition of ϕ(m) via (5.3) immediately shows that ϕu (IA ) = 0 for any Borel set
(m)
(m)
A ⊆ [0, u]; hence we have ϕu (I(0,t] ) = ϕu (I(0,t] )I{u<t} and therefore
(5.4)
(m)
Bt
=
ZT
ϕu(m) (I(0,t] )I{u<t}
dSuTm
0
Zt−
¡ (m)
¢ Tm
Tm
= ϕ(m)
(I(0,t] ) . St−
.
u (I(0,t] ) dSu = ϕ
0
The discussion before Lemma 5.1 now shows that we may apply to ϕ(m) the result in Theorem
4.9, since all its assumptions are satisfied, and therefore we get
(5.5)
giving that
¡
¢
¡
¢
ϕ(m) (I(0,t] ) . SuTm = ϕ(m) . SuTm (I(0,t] )
(m)
Bt
for all u ∈ [0, T ],
¡
m¢
= ϕ(m) . STT (I(0,t] ).
Because ϕ(m) . S Tm has by construction values in IMT , the last expression shows that the
(m)
process B (m) = (Bt )0≤t≤T is RCLL and of finite variation with respect to t. To see that it
¢
¡
Tm
(I(0,s] ) on [0, ∞) × Ω̄ is
is also predictable, note first that the process V (s, t) := ϕ(m) . St−
¡ (m) Tm ¢
well defined, since t 7→ ϕ . St (I(0,s] ) is a martingale by Theorem 4.9 and hence RCLL
with respect to t. Moreover, V (s, t) is (B[0, T ] ⊗ P)-measurable when restricted to s ∈ [0, T ],
since it is right-continuous in s, and left-continuous in t and adapted to IF . Finally, (5.5) also
yields that
(m)
Bt
¡
¢ Tm ¡ (m) T m ¢
= ϕ(m) (I(0,t] ) . St−
= ϕ . St− (I(0,t] ) = V (t, t);
so B (m) is the diagonal of V , and we conclude as in Lemma 3.5 that B (m) is indeed predictable.
This ends the proof.
q.e.d.
The main result of this section is now the following explicit decomposition of the process
of optimal wealths.
Theorem 5.2. Suppose that the IRd -valued semimartingale S satisfies (2.9), and U satisfies
e from Theorem 2.14 of the process X
b of
(2.11)–(2.13). Then the semimartingale version X
optimal wealths admits a decomposition as
(5.6)
e =x+
X
Z
ϑ̄ dS + B̄,
where ϑ̄ is an IRd -valued predictable S-integrable process and B̄ is a predictable RCLL process
of finite variation and null at 0. More explicitly, ϑ̄ is the diagonal
ϑ̄u = ϑ̂(u, u)
52
for u ≥ 0
of ϑ̂, and B̄ is given by
B̄t =
(5.7)
Zt−
¡
0
¢
ϑ̂(t, u) − ϑ̂(u, u) dSu
for t ≥ 0.
The decomposition (5.6) is unique.
Proof. 1) With the familiar stopping times Tm from Lemma 3.1, we have from (5.1), (5.2)
and Lemma 5.1 that
etTm
X
with
(m)
Bt
=
Zt
0
=
Zt
0
=
(m)
ϑ̂(u, u) dSu + Bt
0
¡
¢
ϑ(m) (t, u) − ϑ(m) (u, u) dSuTm
¡
¢
ϑ̂(t ∧ Tm , u ∧ Tm ) − ϑ̂(u ∧ Tm , u ∧ Tm ) I{u≤Tm } dSu
t∧T
Z m
¡
0
=
=x+
t∧T
Z m
¢
ϑ̂(t, u) − ϑ̂(u, u) dSu
(t−)∧T
Z m
0
¡
¢
ϑ̂(t, u) − ϑ̂(u, u) dSu ,
where the last equality follows from (5.4) in the proof of Lemma 5.1. This gives (5.6) as well
as (5.7).
2) In general, if Z > 0 is a local P -martingale with Z0 = 1, then also Z− > 0 and
Z = E(N ) for some local P -martingale N null at 0. For any semimartingale Y , the product
rule and Z = 1 + Z− . N yield
ZY = Y− . Z + Z− . Y + [Z, Y ] = Y− . Z + Z− . (Y + [Y, N ]).
Because Z− > 0, (2.6) therefore implies that ZY is a P -σ-martingale if and only if Y + [Y, N ]
is a P -σ-martingale.
3) To prove uniqueness of the decomposition (5.6), suppose by taking differences that
ϑ . S + C ≡ 0 for some IRd -valued predictable S-integrable ϑ and some predictable RCLL
process C of finite variation. Then ϑ . S = −C which is locally bounded since it is predictable
and RCLL. Thanks to (2.9), we can take some Z ∈ De,σ (S, P ) and write it as Z = E(N ). Since
ZS is a P -σ-martingale, so is S + [S, N ] by step 2), hence also ϑ . (S + [S, N ]) = ϑ . S + [ϑ . S, N ]
53
¡
¢
by (2.6) for any ϑ which is S-integrable because such a ϑ is then also (S +[S, N ])-integrable .
Again by step 2), this implies that the product Z(ϑ . S) is a P -σ-martingale, and this is also
locally P -integrable, because Z is so as a local P -martingale and ϑ . S is locally bounded. Thus
Z(ϑ . S) is a special semimartingale and therefore by Corollary 3.1 of Kallsen (2003) a local
P -martingale, which means that ZC is a local P -martingale. But ZC = Z− . C+C− . Z+[Z, C]
by the product rule, and the last two terms are local P -martingales like Z; for [Z, C], this is
¡
¢
due to Yoeurp’s lemma see Theorem VII.36 in Dellacherie/Meyer (1982) . Because Z− > 0,
we conclude that C is a local P -martingale; but then C ≡ C0 = 0 since C is predictable and
of finite variation, and thus also ϑ . S ≡ 0.
q.e.d.
A. Appendix
This section contains a proof which has been omitted from the main body of the paper to
facilitate a more streamlined presentation.
Proof of Proposition 2.7. a) Since S is a semimartingale and hence RCLL, we have
d
P
[S i ]T < ∞ P -a.s. So setting
sup |St |2 < ∞ P -a.s. and sup |∆St |2 ≤
0≤t≤T
(A.1)
0<t≤T
i=1
¶
µ
dQ
2
2
2
:= const. exp −W − sup |∆St | − sup |St |
dP
0<t≤T
0≤t≤T
clearly defines a probability measure Q ≈ P with bounded density and (2.10). Moreover,
Ze,σ (S T , P ) ⊇ Ze,σ (S, P ) 6= ∅ by (2.9), and hence Ze,σ (S T , Q) 6= ∅ by Theorem 2.6.
b) To prove the second half of 1), we need some results and notations from Choulli/
Stricker (2006); see also Sections III.2 and III.3 in Jacod/Shiryaev (2003). Denote by µ the
random measure associated to the jumps of the (P - and Q-)semimartingale S and represent
its respective compensators under P and Q as
ν(ω, dt, dx) = Ft (ω, dx) dAt (ω),
ν Q (ω, dt, dx) = FtQ (ω, dx) dAt (ω).
£ ¯ ¤
¯
If we denote by ZtQ := EP dQ
dP Ft , 0 ≤ t ≤ T , the density process of Q with respect to P ,
Theorem III.3.24 of Jacod/Shiryaev (2003) gives us the kernel F Q in terms of F by
µ Q¯ ¶
¯
Q
P Z ¯
Ft (ω, dx) = Mµ
P̃ (ω, t, x) Ft (ω, dx).
Q ¯
Z−
Writing c for the constant in the definition (A.1) of Q, we get
µ Q¯ ¶
¯
2
c
P Z ¯
e−|x|
P̃ (ω, t, x) ≤ Q
(A.2)
Mµ
Q ¯
Z−
Zt− (ω)
54
in view of (A.1), the definition of MµP and the fact that W ≥ 0.
By Theorem 3.3 of Choulli/Stricker (2006), a sufficient condition for the existence of the
minimal entropy-Hellinger σ-martingale density Ze for (S T , Q) is that for every λ ∈ IRd ,
Z
|x|eλ
tr
x
FtQ (ω, dx) < ∞
P -a.s. for every t ≤ T .
{|x|>1}
¡
To be accurate, the results in Choulli/Stricker (2006) are given for local (instead of σ-)
martingale densities for S, which means that ZS is required to be a local (instead of a σ-)
martingale under P . However, all the theory extends in a straightforward way from local to
σ-martingales; in fact, it becomes even simpler since one has fewer integrability conditions
¢
to check. For more details, see Section III.6e in Jacod/Shiryaev (2003), or Kallsen (2003).
Using the above expression for F Q and the estimate (A.2), we obtain for every λ ∈ IRd that
Z
|x|e
λtr x
FtQ (ω, dx)
≤
{|x|>1}
≤
c
Q
Zt−
(ω)
c
Q
Zt−
(ω)
Z
|x|eλ
tr
x −|x|2
e
{|x|>1}
exp
µ
(1 + |λ|)2
4
¶
Ft (ω, dx)
Z
Ft (ω, dx) < ∞
{|x|>1}
since Q ≈ P . This completes the proof of part 1) of Proposition 2.7.
e Q) for (S T , Q) exists and is locally
c) Thanks to 1), the entropy-Hellinger process hE (Z,
Q-integrable since it is predictable and RCLL. So take as (Tm ) a localising sequence which also
e Q) ∈ L1 (Q)
e Then each ZeTm is a Q-uniformly integrable Q-martingale, and hE (Z,
reduces Z.
Tm
implies that
¤
£
EQ ZeTm log ZeTm < ∞
(A.3)
by Proposition 3.6 of Choulli/Stricker (2005). (The latter result also holds without the
assumptions that IF is quasi-left-continuous and S is bounded and quasi-left-continuous, as
can easily be seen from its proof.) Setting dQm := ZeTm dQ therefore defines for each m
a probability measure Qm ≈ Q ≈ P such that S Tm is a Qm -σ-martingale. But moreover,
combining (A.3) with Young’s inequality xy ≤ ex + y log y − y for any x, y ≥ 0 shows that
for any random variable X ≥ 0, EQ [eX ] < ∞ implies that EQm [X] < ∞. So (2.10) gives
W ∈ L2 (Qm ) and
sup |St |2 ∈ L1 (Qm ), and so S Tm is indeed a Qm -square-integrable
0≤t≤T
Qm -martingale. This ends the proof.
q.e.d.
Acknowledgments. Tahir Choulli thanks Freddy Delbaen and Martin Schweizer for their
invitation to spend a sabbatical year at ETH Zurich where a large part of this research was
55
done. Their hospitality and warm welcome to the research group in mathematical finance at
the Department of Mathematics in ETH Zurich is greatly appreciated. Financial support by
FIM (the Institute for Mathematical Research at ETH Zurich), by Credit Suisse (through the
funds of Freddy Delbaen) and by NSERC (the Natural Sciences and Engineering Research
Council of Canada, Grant G121210818) is gratefully acknowledged.
Martin Schweizer gratefully acknowledges financial support by the National Centre of
Competence in Research “Financial Valuation and Risk Management” (NCCR FINRISK),
Project D1 (Mathematical Methods in Financial Risk Management). The NCCR FINRISK
is a research instrument of the Swiss National Science Foundation.
Finally, helpful comments and criticisms by several people at several conferences and
workshops have helped us to improve the presentation, and we thank them all for that.
References
S. Ankirchner (2005), “Information and Semimartingales”, PhD thesis, HU Berlin,
http://www.math.hu-berlin.de/∼ankirchn/information.pdf
S. Ankirchner and P. Imkeller (2005), “Finite utility on financial markets with asymmetric information and structure properties of the price dynamics”, Annales de l’Institut Henri
Poincaré B 41, 479–503
J.-P. Ansel and C. Stricker (1994), “Couverture des actifs contingents et prix maximum”,
Annales de l’Institut Henri Poincaré B 30, 303–315
A. Basse and J. Pedersen (2009), “Lévy driven moving averages and semimartingales”,
Stochastic Processes and their Applications 119, 2970–2991
F. Biagini and B. Øksendal (2005), “A general stochastic calculus approach to insider
trading”, Applied Mathematics and Optimization 52, 167–181
K. Bichteler and S. Lin (1995), “On the stochastic Fubini theorem”, Stochastics 54,
271–279
T. Björk, G. Di Masi, Yu. M. Kabanov and W. J. Runggaldier (1997), “Towards a general
theory of bond markets”, Finance and Stochastics 1, 141–174
T. Choulli and M. Schweizer (2010), “Stability of σ-martingale densities in L log L under
a bounded change of measure”, working paper [under construction], ETH Zürich
T. Choulli and C. Stricker (2005), “Minimal entropy-Hellinger martingale measure in
incomplete markets”, Mathematical Finance 15, 465–490
56
T. Choulli and C. Stricker (2006), “More on minimal entropy-Hellinger martingale measure”, Mathematical Finance 16, 1–19
T. Choulli, C. Stricker and J. Li (2007), “Minimal Hellinger martingale measures of order
q”, Finance and Stochastics 11, 399–427
F. Delbaen and W. Schachermayer (1994), “A general version of the fundamental theorem
of asset pricing”, Mathematische Annalen 300, 463–520
F. Delbaen and W. Schachermayer (1998), “The fundamental theorem of asset pricing
for unbounded stochastic processes”, Mathematische Annalen 312, 215–250
F. Delbaen and W. Schachermayer (2006), “The Mathematics of Arbitrage”, Springer
C. Dellacherie and P.-A. Meyer (1978), “Probabilities and Potential”, North-Holland,
Amsterdam
C. Dellacherie and P.-A. Meyer (1982), “Probabilities and Potential B. Theory of Martingales”, North-Holland, Amsterdam
H. Föllmer and A. Schied (2004), “Stochastic Finance: An Introduction in Discrete
Time”, second revised and extended edition, de Gruyter
T. Goll and J. Kallsen (2003), “A complete explicit solution to the log-optimal portfolio
problem”, Annals of Applied Probability 13, 774–799
C. Gollier and R. J. Zeckhauser (2002), “Horizon length and portfolio risk”, Journal of
Risk and Uncertainty 24, 195–212
N. H. Hakansson (1969), “Optimal investment and consumption strategies under risk,
an uncertain lifetime, and insurance”, International Economic Review 10, 443–466
V. Henderson and D. G. Hobson (2007), “Horizon-unbiased utility functions”, Stochastic
Processes and their Applications 117, 1621–1641
T. Hida and M. Hitsuda (1993), “Gaussian Processes”, Translations of Mathematical
Monographs, Volume 120, American Mathematical Society
J. Jacod (1979), “Calcul Stochastique et Problèmes de Martingales”, Lecture Notes in
Mathematics 714, Springer
J. Jacod and A. N. Shiryaev (2003), “Limit Theorems for Stochastic Processes”, second
edition, Springer
S. Jaggia and S. Thosar (2000), “Risk aversion and the investment horizon: A new
perspective on the time diversification debate”, Journal of Behavioral Finance 1, 211–215
57
Yu. M. Kabanov (1997), “On the FTAP of Kreps–Delbaen–Schachermayer”, in: Yu. M.
Kabanov, B. L. Rozovskii and A. N. Shiryaev (eds.), ‘Statistics and Control of Stochastic
Processes. The Liptser Festschrift’, World Scientific, 191–203
J. Kallsen (2003), “σ-localization and σ-martingales”, Theory of Probability and its
Applications 48, 152–163
I. Karatzas and C. Kardaras (2007), “The numéraire portfolio in semimartingale financial
models”, Finance and Stochastics 11, 447–493
I. Karatzas and G. Žitković (2003), “Optimal consumption from investment and random
endowment in incomplete semimartingale markets”, Annals of Probability 31, 1821–1858
D. O. Kramkov (1996), “Optional decomposition of supermartingales and hedging contingent claims in incomplete security markets”, Probability Theory and Related Fields 105,
459–479
D. Kramkov and W. Schachermayer (1999), “The asymptotic elasticity of utility functions and optimal investment in incomplete markets”, Annals of Applied Probability 9, 904–
950
D. Kramkov and M. Sı̂rbu (2006), “On the two-times differentiability of the value functions in the problem of optimal investment in incomplete markets”, Annals of Applied Probability 16, 1352–1384
K. Larsen and G. Žitković (2008), “On the semimartingale property via bounded logarithmic utility”, Annals of Finance 4, 255–268
A. Lazrak (1998), “Optimal martingale restrictions and horizon effect on portfolio selection”, Prépublications de l’Equipe d’Analyse et Probabilités, No. 91, Université d’Évry,
http://www.maths.univ-evry.fr/prepubli/index.html
V. A. Lebedev (1995), “The Fubini theorem for stochastic integrals with respect to
L -valued random measures depending on a parameter”, Theory of Probability and its Applications 40, 285–293
0
P. Protter (1985), “Volterra equations driven by semimartingales”, Annals of Probability
13, 519–530
P. Protter (2005), “Stochastic Integration and Differential Equations. A New Approach”,
second edition, version 2.1 (corrected 3rd printing), Springer
N. Siebenmorgen and M. Weber (2004), “The influence of different investment horizons
on risk behavior”, Journal of Behavioral Finance 5/2, 75–90
58
M. E. Yaari (1965), “Uncertain lifetime, life insurance, and the theory of the consumer”,
Review of Economic Studies 32, 137–150
T. Zariphopoulou and G. Žitković (2008), “Maturity-independent risk measures”, preprint, University of Texas at Austin, http://arxiv.org/abs/0710.3892v2
59