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arXiv:1102.2050v1 [math.PR] 10 Feb 2011
On stochastic calculus related to financial assets
without semimartingales
Rosanna COVIELLO1, Cristina DI GIROLAMI2and Francesco RUSSO3,4
1HSBC, 103, av. des Champs-Elysées, F-75419 Paris Cedex 09 (France).
2LUISS Guido Carli - Libera Università Internazionale degli Studi Sociali Guido Carli di Roma (Italy).
3ENSTA ParisTech, Unité de Mathématiques appliquées, 32, Boulevard Victor, F-75739 Paris Cedex 15 (France).
4INRIA Rocquencourt and Cermics Ecole des Ponts, Projet MATHFI. Domaine de Voluceau, BP 105 F-78153 Le
Chesnay Cedex (France).
Abstract This paper does not suppose a priori that the evolution of the price of a financial
asset is a semimartingale. Since possible strategies of investors are self-financing, previous
prices are forced to be finite quadratic variation processes. The non-arbitrage property is
not excluded if the class A of admissible strategies is restricted. The classical notion of
martingale is replaced with the notion of A-martingale. A calculus related to A-martingales
with some examples is developed. Some applications to no-arbitrage, viability, hedging and
the maximization of the utility of an insider are expanded. We finally revisit some no arbitrage
conditions of Bender-Sottinen-Valkeila type.
2010 MSC: 60G48, 60H05, 60H07, 60H10, 91B16, 91B24, 91B70.
Keywords and phrases: A-martingale, weak k-order Brownian motion, no-semimartingale,
utility maximization, insider, no-arbitrage, viability, hedging.
1. Introduction
This article is devoted to the memory of Professor Paul Malliavin, a legend in mathematics. Among
his huge and fruitful contributions there is the celebrated Malliavin calculus. Malliavin calculus
was succesfully applied to many areas in probability and analysis but also in financial mathematics.
Prof. Malliavin himself in the last part of his career was very productive in this field, as the excellent
monograph [30] written with Prof. Thalmaier shows. Especially the third named author is grateful
for all the mathematical interactions he could have with him. Prof. Malliavin was actively present
to a talk of F. Russo which included the first part of the topic of the present paper.
According to the fundamental theorem of asset pricing of Delbaen and Schachermayer in [13],
Chapter 14, in absence of free lunches with vanishing risk (NFLVR), when investing possibilities
run only through simple predictable strategies with respect to some filtration G, the price process
of the risky asset S is forced to be a semimartingale. However (NFLVR) condition could not be
reasonable in several situations. In that case S may not be a semimartingale. We illustrate here
some of those circumstances.
Generally, admissible strategies are let vary in a quite large class of predictable processes with
respect to some filtration G, representing the information flow available to the investor. As a matter
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of fact, the class of admissible strategies could be reduced because of different market regulations
or for practical reasons. For instance, the investor could not be allowed to hold more than a
certain number of stock shares. On the other hand it could be realistic to impose a minimal delay
between two possible transactions as suggested by Cheridito ([9]), see also [24]: when the logarithmic
price log(S) is a geometric fractional Brownian motion (fbm), it is impossible to realize arbitrage
possibilities satisfying that minimal requirement. We remind that without that restriction, the
market admits arbitrages, see for instance [36, 46, 43]. When the logarithmic price of S is a
geometric fbm or some particular strong Markov process, arbitrages can be excluded taking into
account proportional transactions costs: Guasoni ([22]) has shown that, in that case, the class
of admissible strategies has to be restricted to bounded variation processes and this rules out
arbitrages.
Besides the restriction of the class of admissible strategies, the adoption of non-semimartingale
models finds its justification when the no-arbitrage condition itself is not likely.
Empirical observations reveal, indeed, that S could fail to be a semimartingale because of market
imperfections due to micro-structure noise, as intra-day effects. A model which considers those
imperfections would add to W, the Brownian motion describing log-prices, a zero quadratic varia-
tion process, as a fractional Brownian motion of Hurst index greater than1
Theoretically arbitrages in very small time interval could be possible, which would be compatible
with the lack of semimartingale property.
2, see for instance [48].
At the same way if (FLVR) are not possible for an honest investor, an inside trader could realize a
free lunch with respect to the enlarged filtration G including the one generated by prices and the
extra-information. Again in that case S may not be a semimartingale. The literature concerning
inside trading and asymmetry of information has been extensively enriched by several papers
in the last ten years; among them we quote Pikowski and Karatzas ([33]), Grorud and Pontier
([21]), Amendinger, Imkeller and Schweizer ([1]). They adopt enlargement of filtration techniques
to describe the evolution of stock prices in the insider filtration.
Recently, some authors approached the problem in a new way using in particular forward integrals,
in the framework of stochastic calculus via regularizations. For a comprehensive survey of that
calculus see [42]. Indeed, forward integrals could exist also for non-semimartingale integrators.
Leon, Navarro and Nualart in [28], for instance, solve the problem of maximization of expected
logarithmic utility of an agent who holds an initial information depending on the future of prices.
They operate under technical conditions which, a priori, do not imply the classical Assumption (H’)
for enlargement considered in [26]. Using forward integrals, they determine the utility maximum.
However, a posteriori, they found out that their conditions oblige S to be a semimartingale.
Biagini and Øksendal ([5]) considered somehow the converse implication. Supposing that the max-
imum utility is attained, they proved that S is a semimartingale. Ankirchner and Imkeller ([2])
continue to develop the enlargement of filtrations techniques and show, among other thinks, a
similar result as [5] using the fundamental theorem of asset pricing of Delbaen-Schachermayer. In
particular they establish a link between that fundamental theorem and finite utility.
In our paper we treat a market where there are one risky asset, whose price is a strictly positive
process S, and a less risky asset with price S0, possibly riskless but a priori only with bounded
variation. A class A of admissible trading strategies is specified. If A is not large enough to generate
all predictable simple strategies, then S has no need to be a semimartingale, even requiring the
absence of free lunches among those strategies.
The aim of the present paper is to settle the basis of a fundamental (even though preliminary)
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calculus which, in principle, allows to model financial assets without semimartingales. Of course
this constitutes the first step of a more involved theory generalizing the classical theory related to
semimartingales. The objective is two-fold.
1. To provide a mathematical framework which extends Itô calculus conserving some particular
aspects of it in a non-semimartingale framework. This has an interest in itself, independently
from mathematical finance. The two major tools are forward integrals and A-martingales.
2. To build the basis of a corresponding financial theory which allows to deal with several
problems as hedging and non-arbitrage pricing, viability and completeness as well as with
utility maximization.
For the sake of simplicity in this introduction we suppose that the less risky asset S0is constant
and equal to 1.
As anticipated, a natural tool to describe the self-financing condition is the forward integral of
an integrand process Y with respect to an integrator X, denoted by?t
F = G1; G represents the flow of information available to the investor. A self-financing portfolio
is a pair (X0,h) where X0is the initial value of the portfolio and h is a G-adapted and S-forward
integrable process specifying the number of shares of S held in the portfolio. The market value
process X of such a portfolio, is given by X0+?·
This formulation of self-financing condition is coherent with the case of transactions at fixed discrete
dates. Indeed, let we consider a buy-and-hold strategy, i.e. a pair (X0,h) with h = ηI(t0,t1],0 ≤ t0≤
t1≤ 1, and η being a Gt0-measurable random variable. Using the definition of forward integral it
is not difficult to see that: Xt0= X0, Xt1= X0+ η(St1− St0). This implies h0
h0
0Y d−X; see section 2 for
definitions. Let G = (Gt)0≤t≤1be a filtration on an underlying probability space (Ω,F,P), with
0hsd−Ss, while h0
t= Xt− Sthtconstitutes the
number of shares of the less risky asset held.
t0+= X0− ηSt0,
t1+= X0+ η(St1− St0) and
Xt0= ht0+St0+ h0
t0+,Xt1= ht1+St1+ h0
t1+:
(1)
at the re-balancing dates t0 and t1, the value of the old portfolio must be reinvested to build
the new portfolio without exogenous withdrawal of money. By ht+, we denote lims↓ths. The use
of forward integral or other pathwise type integral is crucial. Previously some other functional
integrals as Skorohod type integrals, involving Wick products see for instance [4]. They are however
not economically so appropriated as for instance [7] points out.
In this paper A will be a real linear subspace of all self-financing portfolios and it will constitute,
by definition, the class of all admissible portfolios. A will depend on the kind of problems one
has to face: hedging, utility maximization, modeling inside trading. If we require that S belongs to
A, then the process S is forced to be a finite quadratic variation process. In fact,?·
?·
However, there could be situations in which S may be allowed not to have finite quadratic variation.
In fact a process h could be theoretically an integrand of a process S without finite quadratic
variation if it has for instance bounded variation.
0Sd−S exists
if and only if the quadratic variation [S] exists, see [42]; in particular one would have
0
Ssd−Ss=1
2(S2− S2
0− [S]).
Even if the price process (St) is an (Ft)-adapted process, the class A is first of all a class of
integrands of S. We recall the significant result of [38] Proposition 1.2. Whenever A includes the
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class of bounded (Ft)-processes then S is forced to be a semimartingale. In general, the class of
forward integrands with respect to S could be much different from the set of locally bounded
predictable (Ft)-processes.
A crucial concept is provided by A-martingale processes. Those processes naturally intervene in
utility maximization, arbitrage and uniqueness of hedging prices.
A process M is said to be an A-martingale if for any process Y ∈ A,
E
??·
0
Y d−M
?
= 0.
If for some filtration F with respect to which M is adapted, A contains the class of all bounded
F-predictable processes, then M is an F-martingale.
L will be the sub-linear space of L0(Ω) representing a set of contingent claims of interest for
one investor. An A-attainable contingent claim will be a random variable C for which there is
a self-financing portfolio (X0,h) with h ∈ A and
?1
X0will be called replication price for C.
A portfolio (X0,h) is said to be an A-arbitrage if h ∈ A, X1≥ X0almost surely and P{X1−X0>
0} > 0. We denote by M the set of probability measures being equivalent to the initial probability
P under which S is an A-martingale. If M is non empty then the market is A-arbitrage free. In
fact if Q ∈ M, given a pair (X0,h) which is an A-arbitrage, then EQ[X1−X0] = EQ[?1
the process hη, for any bounded random variable η in G0and h in A, still belongs to A. Moreover
X0= EQ[C|G0]. In reality, under the weaker assumption that the market is A-arbitrage free, the
replication price is still unique, see Proposition 4.26.
C = X0+
0
hsd−Ss.
0hd−S] = 0.
In that case the replication price X0of an A-attainable contingent claim C is unique, provided that
Using the inspiration coming from [3], we reformulate a non-arbitrage property related to an
underlying S which verifies the so-called full support condition. We provide some theorems which
generalize some aspects of [3]. Let us denote by Ss(·) the history at time s of process S. If S is has
finite quadratic variation and [S]t=?t
excluding arbitrage opportunities. See Proposition 4.40, Example 4.41 and the central Theorem
4.43. However, since there are many examples of non-semimartingale processes S fulfilling the full
support conditions, the class A will not generate the canonical filtration of S.
The market will be said (A,L)-attainable if every element of L is A-attainable. If the market is
(A,L)-attainable then all the probabilities measures in M coincide on σ(L), see Proposition 4.27.
If σ(L) = F then M is a singleton: this result recovers the classical case, i.e. there is a unique
probability measure under which S is a semimartingale.
0σ2(s,Ss(·))S2
sds,t ∈ [0,T] and σ : [0,T] × C([−1,0] → R
is continuous, bounded and non-degenerate. It is possible to provide rich classes A of strategies
In these introductory lines we will focus on one particular toy model.
For simplicity we illustrate the case where [log(S)]t = σ2t,
all European contingent claims C = ψ(S1) where ψ is continuous with polynomial growth. We
consider the case A = AS, where
AS
with polynomial growth}.
σ > 0. We choose as L the set of
={(u(t,St)),0 ≤ t < 1| u : [0,1] × R → R, Borel-measurable
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If the corresponding M is non empty and A = AS, as assumed in this section, the law of Sthas
to be equivalent to Lebesgue measure for every 0 < t ≤ 1, see Proposition 4.21.
An example of A-martingale is the so called weak Brownian motion of order k = 1 and
quadratic variation equal to t. That notion was introduced in [20]: a weak Brownian motion of
order 1 is a process X such that the law of Xtis N(0,t) for any t ≥ 0.
Such a market is (A,L)-attainable: in fact, a random variable C = ψ(S1) is an A-attainable
contingent claim. To build a replicating strategy the investor has to choose v as solution of the
following problem
?
and X0= v(0,S0). This follows easily after application of Itô’s formula contained in Proposition
2.10, see Proposition 4.28. This technique was introduced by [45]. Subsequent papers in that
direction are those of [49] and [3] which shows in particular that several path dependent options
can be covered only assuming that S has the same quadratic variation as geometrical Brownian
motion. In Proposition 4.31 and in Proposition 4.30, we highlight in particular that this method
can be adjusted to hedge also Asian contingent claims and some options only depending on a finite
number of dates of the underlying price. This discussion is continued in [15] and [16] which perform
a suitable infinite dimensional calculus via regularizations, opening the way to the possible hedge
of much reacher classes of path dependent options.
∂tv(t,x) +1
v(1,x)
2σ2x2∂(2)
xxv(t,x)=
=
0
ψ(x)
Given an utility function satisfying usual assumptions, it is possible to show that the maximum
π is attained on a class of portfolios fulfilling conditions related to Assumption 5.8, if and only
if there exists a probability measure under which log(S) −?·
Definition 3.6, then S is a classical semimartingale.
0
?σ2πt−1
2σ2?dt is an A-martingale,
see Proposition 5.15. Therefore if A is big enough to fulfill conditions related to Assumption D in
Before concluding we introduce some examples of motivating pertinent classes A.
1. Transactions at fixed dates. Let 0 = t0< t1< ... < tm= 1 be a fixed subdivision of
[0,1]. The price process is continuous but the transactions take place at the fixed considered
dates. A includes the class of predictable processes of the type
Ht=
n−1
?
i=0
Hti1(ti,ti+1],
where Htiis an Ftimeasurable random variable. A process S such that (Sti) is an (Fti)
-martingale is an A-martingale.
2. Cheridito type strategies. According to [9, 24], that class A of strategies includes bounded
processes H such that the time between two transactions is greater or equal than τ for some
τ > 0.
3. Delay or anticipation. If τ ∈ R, then A is constituted by integrable processes H such that
Htis F(t+τ)+-measurable. A process S which is an (F(t+τ)+) martingale is an A-martingale.
4. Let G be an anticipating random variable with respect to F. A is a class of processes of the
type Ht= h(t,G), h(t,x) is a random field fulfilling some Kolmogorov continuity lemma in
x.
5. Other examples are described in section 4.
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Those considerations show that most of the classical results of basic financial theory admit a natural
extension to non-semimartingale models.
The paper is organized as follows. In section 2, we introduce stochastic calculus via regularizations
for forward integrals. Section 3 considers, a priori, a class A of integrands associated with some
integrator X and focuses the notion of A-martingale with respect to A. We explore the relation
between A-martingales and weak Brownian motion; later we discuss the link between the existence
of a maximum for a an optimization problem and the A-martingale property.
The class A is related to classes of admissible strategies of an investor. The admissibility con-
cern theoretical or financial (regulatory) restrictions. At the theoretical level, classes of admissible
strategies are introduced using Malliavin calculus, substitution formulae and Itô fields. Regarding
finance applications, the class of strategies defined using Malliavin calculus is useful when log(S) is
a geometric Brownian motion with respect to a filtration F contained in G; the use of substitution
formulae naturally appear when trading with an initial extra information, already available at time
0; Itô fields apply whenever S is a generic finite quadratic variation process. Section 4 discusses
some of previous examples and it deals with basic applications to mathematical finance. We define
self-financing portfolio strategies and we provide examples. Technical problems related to the use
of forward integral in order to describe the evolution of the wealth process appear. Those problems
arise because of the lack of chain rule properties. Later, we discuss absence of A-arbitrages, (A,L)-
attainability and hedging. In Section 5 we analyze the problem of maximizing expected utility from
terminal wealth. We obtain results about the existence of an optimal portfolio generalizing those
of [28] and [5].
2. Preliminaries
For the convenience of the reader we give some basic concepts and fundamental results about
stochastic calculus with respect to finite quadratic variation processes which will be extensively
used later. For more details we refer the reader to [42].
In the whole paper (Ω,F,P) will be a fixed probability space. For a stochastic process X =
(Xt,0 ≤ t ≤ 1) defined on (Ω,F,P) we will adopt the convention Xt= X(t∨0)∧1, for t in R. Let
0 ≤ T ≤ 1. We will say that a sequence of processes (Xn
probability (ucp) on [0,T] toward a process (Xt,0 ≤ t ≤ T), if supt∈[0,T]|Xn
to zero in probability.
t,0 ≤ t ≤ T)n∈Nconverges uniformly in
t− Xt| converges
Definition 2.1.
respectively in C0([0,T]) and L1([0,T]). Set, for every 0 ≤ t ≤ T,
I(ε,Y,X,t) =1
1. Let X = (Xt,0 ≤ t ≤ T) and Y = (Yt,0 ≤ t ≤ T) be processes with paths
ε
?t
0
Ys(Xs+ε− Xs)ds,
and
C(ε,X,Y,t) =1
ε
?t
0
(Ys+ε− Ys)(Xs+ε− Xs)ds.
If I(ε,Y,X,t) converges in probability for every t in [0,T], and the limiting process admits a
continuous version I(Y,X,t) on [0,T], Y is said to be X-forward integrable on [0,T]. The
process (I(Y,X,t),0 ≤ t ≤ T) is denoted by?·
0Y d−X. If I(ε,Y,X,·) converges ucp on [0,T]
0Y d−X is the limit ucp of its regularizations.we will say that the forward integral?·
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2. If (C(ε,X,Y,t),0 ≤ t ≤ T) converges ucp on [0,T] when ε tends to zero, the limit will be
called the covariation process between X and Y and it will be denoted by [X,Y ]. If X = Y,
[X,X] is called the finite quadratic variation of X: it will also be denoted by [X], and X
will be said to be a finite quadratic variation process on [0,T].
Definition 2.2. We will say that a process X = (Xt,0 ≤ t ≤ T), is localized by the sequence
?Ωk,Xk?
Remark 2.3. Let (Xt,0 ≤ t ≤ T) and (Y,0 ≤ t ≤ T) be two stochastic processes. The following
statements are true.
1. Let Y and X be localized by the sequences?Ωk,Xk?
on [0,T] and
?·
2. Given a random time T ∈ [0,1] we often denote XT
3. If Y is X-forward integrable on [0,T], then Y I[0,T ]is X-forward integrable for every random
time 0 ≤ T ≤ T, and
?·
4. If the covariation process [X,Y ] exists on [0,T], then the covariation process [XT,YT] exists
for every random time 0 ≤ T ≤ T, and
?XT,YT?= [X,Y ]T.
respectively in C0([0,T]) and L1
0|Ys|ds < +∞ for any t < T.
1. If Y I[0,t]is X-forward integrable for every 0 ≤ t < T, Y is said locally X-forward inte-
grable on [0,T). In this case there exists a continuous process, which coincides, on every
compact interval [0,t] of [0,1), with the forward integral of Y I[0,t]with respect to X. That
process will still be denoted with I(·,Y,X) =?·
2. If Y is locally X-forward integrable and limt→TI(t,Y,X) exists almost surely, Y is said
X-improperly forward integrable on [0,T].
3. If the covariation process [Xt,Yt] exists, for every 0 ≤ t < T, we say that the covariation
process [X,Y ] exists locally on [0,T) and it is still denoted by [X,Y ]. In this case there
exists a continuous process, which coincides, on every compact interval [0,t] of [0,1), with
the covariation process?X,Y I[0,t]
4. If the covariation process [X,Y ] exists locally on [0,T) and limt→T[X,Y ]texists, the limit will
be called the improper covariation process between X and Y and it will still be denoted by
[X,Y ]. If X = Y, [X,X] we will say that the quadratic variation of X exists improperly
on [0,T].
k∈N∗, if P?∪+∞
k=0Ωk
?= 1, Ωh⊆ Ωk, if h ≤ k, and IΩkXk= IΩkX, almost surely for every
k in N.
k∈Nand?Ωk,Yk?
k∈N, respectively, such
that Ykis Xk-forward integrable on [0,T] for every k in N. Then Y is X-forward integrable
0
Y d−X =
?·
0
Ykd−Xk,
on Ωk, a.s..
t = Xt∧T, t ∈ [0,T].
0
YsI[0,t]d−Xs=
?·∧t
0
Ysd−Xs.
Definition 2.4. Let X = (Xt,0 ≤ t ≤ T) and Y = (Yt,0 ≤ t < T) be processes with paths
loc([0,T)), i.e.?t
0Y d−X.
?. That process will still be denoted with [X,Y ]. If X = Y,
[X,X] we will say that the quadratic variation of X exists locally on [0,T].
Remark 2.5. Let X = (Xt,0 ≤ t ≤ T) and Y = (Yt,0 ≤ t ≤ T) be two stochastic processes
whose paths are C0([0,T]) and L1([0,T]), respectively. If Y is X-forward integrable on [0,T] then
its restriction to [0,T) is X-improperly forward integrable and the improper integral coincides with
the forward integral of Y with respect to X.
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Definition 2.6. A vector??X1
In the sequel if T = 1 we will omit to specify that objects defined above exist on the interval [0,1]
(or [0,1), respectively).
t,...,Xm
t
?,0 ≤ t ≤ T?of continuous processes is said to have all its
mutual brackets on [0,T] if?Xi,Xj?exists on [0,T] for every i,j = 1,...,m.
Proposition 2.7. Let M = (Mt,0 ≤ t ≤ T) be a continuous local martingale with respect to some
filtration F = (Ft)t∈[0,T]of F. Then the following properties hold.
1. The process M is a finite quadratic variation process on [0,T] and its quadratic variation
coincides with the classical bracket appearing in the Doob decomposition of M2.
2. Let Y = (Yt,0 ≤ t ≤ T) be an F-adapted process with left continuous and bounded paths. Then
Y is M-forward integrable on [0,T] and
?·
Proposition 2.8. Let V = (Vt,0 ≤ t ≤ T) be a bounded variation process and Y = (Yt,0 ≤ t ≤ T),
be a process with paths being bounded and with at most countable discontinuities. Then the following
properties hold.
1. The process Y is V -forward integrable on [0,T] and
Stieltjes integral denoted with?·
2. The covariation process [Y,V ] exists on [0,T] and it is equal to zero. In particular a bounded
variation process has zero quadratic variation.
?·
0Y d−M coincides with the classical Itô integral
0Y dM.
?·
0Y d−V coincides with the Lebesgue-
0Y dV.
Corollary 2.9. Let X = (Xt,0 ≤ t ≤ T) be a continuous process and Y = (Yt,0 ≤ t ≤ T) a
bounded variation process. Then
?·
Proposition 2.10. Let X = (Xt,0 ≤ t ≤ T) be a continuous finite quadratic variation process
and V = ((V1
t),0 ≤ t ≤ T) be a vector of continuous bounded variation processes. Then
for every u in C1,2(Rm×R), the process (∂xu(Vt,Xt),0 ≤ t ≤ T) is X-forward integrable on [0,T]
and
?·
1
2
0
XY − X0Y0=
0
XsdYs+
?·
0
Ysd−Xs.
t,...,Vm
u(V,X)=
u(V0,X0) +
m
?
i=1
0
∂viu(Vt,Xt)dVi
t+
?·
0
∂xu(Vt,Xt)d−Xt
+
?·
∂(2)
xxu(Vt,Xt)d[X]t.
Proposition 2.11. Let X = (X1
all its mutual brackets. Let ψ : Rm→ R be of class C2(Rm) and Y = ψ(X). Then Z is Y -forward
integrable on [0,T], if and only if Z∂xiψ(X) is Xi-forward integrable on [0,T], for every i = 1,...,m
and
?·
i=1
t,...,Xm
t,0 ≤ t ≤ T) be a vector of continuous processes having
0
Zd−Y =
m
?
?·
0
Z∂xiψ(X)d−Xi+1
2
m
?
i,j=0
?·
0
Z∂(2)
xixjψ(X)d?Xi,Xj?.
Proof. The proof derives from Proposition 4.3 of [41]. The result is a slight modification of that
one. It should only be noted that there forward integral of a process Y with respect to a process
X was defined as limit ucp of its regularizations.
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Remark 2.12. Taking Z = 1, the chain rule property described in Proposition 2.11 implies in
particular the classical Itô formula for finite quadratic variation processes stated for instance in
[39] or in a discretization framework in [19].
3.
A-martingales
Throughout this section A will be a real linear space of measurable processes indexed by [0,1) with
paths which are bounded on each compact interval of [0,1).
We will denote with F = (Ft)t∈[0,1]a filtration indexed by [0,1] and with P(F) the σ-algebra
generated by all left continuous and F-adapted processes. In the remainder of the paper we will
adopt the notations F and P(F) even when the filtration F is indexed by [0,1). At the same way,
if X is a process indexed by [0,1], we shall continue to denote with X its restriction to [0,1).
3.1. Definitions and properties
Definition 3.1. A process X = (Xt,0 ≤ t ≤ 1) is said A-martingale if every θ in A is X-
improperly forward integrable and E
??t
0θsd−Xs
?
= 0 for every 0 ≤ t ≤ 1.
Definition 3.2. A process X = (Xt,0 ≤ t ≤ 1) is said A-semimartingale if it can be written as
the sum of an A-martingale M and a bounded variation process V, with V0= 0.
Remark 3.3.
1. If X is a continuous A-martingale with X belonging to A, its quadratic vari-
ation exists improperly. In fact, if
[X,X] exists improperly and [X,X] = X2− X2
of [41] for details.
?·
0Xtd−Xt exists improperly, it is possible to show that
0− 2?·
0Xsd−Xs. We refer to Proposition 4.1
2. Let X be a continuous square integrable martingale with respect to some filtration F. Suppose
that every process in A is the restriction to [0,1) of a process (θt,0 ≤ t ≤ 1) which is F-
adapted, it has left continuous with right limit paths (cadlag) and E
Then X is an A-martingale.
3. In [20] the authors introduced the notion of weak-martingale. A semimartingale X is a
??t
Borel-measurable. Clearly we can affirm the following. Suppose that A contains all processes
of the form f(·,X), with f as above. Let X be a semimartingale which is an A-martingale.
Then X is a weak-martingale.
??1
0θ2
td[X]t
?
< +∞.
weak-martingale if E
0f(s,Xs)dXs
?
= 0, 0 ≤ t ≤ 1, for every f : R+× R → R, bounded
Proposition 3.4. Let X be a continuous A-martingale. The following statements hold true.
1. If X belongs to A, X0= 0 and [X,X] = 0. Then X ≡ 0.
2. Suppose that A contains all bounded P(F)-measurable processes. Then X is an F-martingale.
Proof. From point 1. of Remark 3.3, E?X2
Regarding point 2. it is sufficient to observe that processes of type IAI(s,t], with 0 ≤ s ≤ t ≤ 1, and
A in Fsbelong to A. Moreover?1
t
?= 0, for all 0 ≤ t ≤ 1.
0IAI(s,t](r)d−Xr= IA(Xt−Xs). This imply E[Xt−Xs| Fs] = 0,
0 ≤ s ≤ t ≤ 1.
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Corollary 3.5. The decomposition of an A-semimartingale X in definition 3.2 is unique among
the class of processes of type M + V, being M a continuous A-martingale in A and V a bounded
variation process.
Proof. If M +V and N +W are two decompositions of that type, then M −N is a continuous A-
martingale in A starting at zero with zero quadratic variation. Point 1. of Proposition 3.4 permits
to conclude.
The following Proposition gives sufficient conditions for an A-martingale to be a martingale with
respect to some filtration F, when A is made up of P(F)-measurable processes. It constitutes a
generalization of point 2. in Proposition 3.4.
Definition 3.6. We will say that A satisfies Assumption D with respect to a filtration F if
1. Every θ in A is F-adapted;
2. For every 0 ≤ s < 1 there exists a basis Bsfor Fs, with the following property. For every A
in Bsthere exists a sequence of Fs-measurable random variables (Θn)n∈N, such that for each
n the process ΘnI[0,1)belongs to A, supn∈N|Θn| ≤ 1, almost surely and
lim
n→+∞Θn= IA, a.s.
Proposition 3.7. Let X = (Xt,0 ≤ t ≤ 1) be a continuous A-martingale adapted to some filtration
F, with Xt belonging to L1(Ω) for every 0 ≤ t ≤ 1. Suppose that A satisfies Assumption D with
respect to F. Then X is an F-martingale.
Proof. We have to show that for all 0 ≤ s ≤ t ≤ 1, E[IA(Xt− Xs)] = 0, for all A in Fs. We fix
0 ≤ s < t ≤ 1 and A in Bs. Let (Θn)n∈N be a sequence of random variables converging almost
surely to IAas in the hypothesis. Since X is an A-martingale, E [Θn(Xt− Xs)] = 0, for all n in
N. We note that Xt− Xsbelongs to L1(Ω), then, by Lebesgue dominated convergence theorem,
|E[IA(Xt− Xs)]| ≤lim
n→+∞E[|IA− Θn||Xt− Xs|] = 0.
Previous result extends to the whole σ-algebra Fs and this permits to achieve the end of the
proof.
Some interesting properties can be derived taking inspiration from [20].
For a process X, we will denote
AX
={(ψ(t,Xt)),0 ≤ t < 1| ψ : [0,1] × R → R, Borel-measurable
with polynomial growth }.
(2)
Proposition 3.8. Let X be a continuous A-martingale with A = AX.
Then, for every ψ in C2(R) with bounded first and second derivatives, the process
ψ(X) −1
2
?·
0
ψ′′(Xs)d[X,X]s
is an A-martingale.
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Proof. The process X belongs to A. In particular, X admits improper quadratic variation. We set
Y = ψ(X) −1
?t
Since θψ′(X) still belongs to A, θ is Y -improperly forward integrable and
?·
We conclude taking the expectation in equality (3).
2
?·
0ψ′′(Xs)d[X,X]s. Let θ in AX. By Proposition 2.11, for every 0 ≤ t < 1
?t
0
θsd−Ys=
0
θsψ′(Xs)d−Xs.
0
θtd−Yt=
?·
0
θtψ′(Xt)d−Xt.
(3)
Proposition 3.9. Suppose that A is an algebra. Let X and Y be two continuous A-martingales
with X and Y in A.
Then the process XY − [X,Y ] is an A-martingale .
Proof. Since A is a real linear space, (X+Y ) belongs to A. In particular by point 1. of Remark 3.3,
[X + Y,X + Y ], [X,X] and [Y,Y ] exist improperly. This implies that [X,Y ] exists improperly too
and that it is a bounded variation process. Therefore the vector (X,Y ) admits all its mutual brack-
ets on each compact set of [0,1). Let θ be in A. Since A is an algebra, θX and θY belong to A and
so both?·
?·
Taking the expectation in the last expression we then get the result.
0θsXsd−Ys and?·
0θsYsd−Xslocally exist. By Proposition 2.11?·
?·
0θtd−(XtYt− [X,Y ])
exists improperly too and
0
θtd−(XtYt− [X,Y ]t) =
0
Ytθtd−Xt+
?·
0
Xtθtd−Yt.
We recall a notion and a related result of [10].
A process R is strongly predictable with respect to a filtration F, if
∃ δ > 0, such that Rε+·is F-adapted, for every ε ≤ δ.
Proposition 3.10. Let R be an F-strongly predictable continuous process. Then for every contin-
uous F-local martingale Y, [R,Y ] = 0.
Proposition 3.10 combined with Proposition 3.9 implies Proposition 3.11 and Corollary 3.12.
Proposition 3.11. Let A, X and Y be as in Proposition 3.9. Assume, moreover, that X is an F-
local martingale, and that Y is strongly predictable with respect to F. Then XY is an A-martingale.
Corollary 3.12. Let A, X and Y be as in Proposition 3.9. Assume that X is a local martingale
with respect to some filtration G and that Y is either G-independent, or G0-measurable. Then XY
is an A-martingale.
Proof. If Y is G-independent, it is sufficient to apply previous Proposition with
F =??
ε>0Gt+ε∨ σ(Y )?
t∈[0,1]. Otherwise one takes F = G.
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3.2. A-martingales and Weak Brownian motion
We proceed defining and discussing processes which are weak-Brownian motions in order to exhibit
explicit examples of A-martingales.
Definition 3.13. ([20]) A stochastic process (Xt,0 ≤ t ≤ 1) is a weak Brownian motion of
order k if for every k-tuple (t1,t2,...,tk)
(Xt1,Xt2,...,Xtk)law
= (Wt1,Wt2,...,Wtk)
where (Wt,0 ≤ t ≤ 1) is a Brownian motion.
Remark 3.14.
1. Using the definition of quadratic variation it is not difficult to show for a
weak Brownian motion of order k ≥ 4, we have [X]t= t.
2. In [20] it is shown that for any k ≥ 1, there exists a weak k-order Brownian motion which is
different from classical Wiener process.
3. If k ≥ 2 then X admits a continuous modification and can be therefore always considered
continuous.
For a process (Xt,0 ≤ t ≤ 1), we set
A1
X
={(ψ(t,Xt),0 ≤ t ≤ 1, with polynomial growth s.t ψ = ∂xΨ
Ψ ∈ C1,2([0,1] × R) with ∂tΨ and ∂(2)
xxΨ bounded.
?
.
Assumption 3.15. Let σ : [0,1]×R → R be a Borel-measurable and bounded function. We suppose
moreover that the equation
?
∂tνt(dx) =1
ν0(dx) = δ0.
2∂(2)
xx
?σ2(t,x)νt(dx)?
(4)
admits a unique solution (νt)t∈[0,1]in the sense of distributions, in the class of continuous functions
t ?→ M(R) where M(R) is the linear space of finite signed Borel real measures, equipped with the
weak topology.
Remark 3.16.
and, in that case, νt= N(0,σ2t), for every 0 ≤ t ≤ 1.
2. Suppose moreover the following. For every compact set of [0,1] × R σ is lower bounded by a
positive constant. We say in this case that σ is non-degenerate.
Exercises 7.3.2-7.3.4 of [47] (see also [27], Refinements 4.32, Chap. 5) say that there is a
weak unique solution to equation dZ = σ(·,Z)dW,Z0= 0, W being a classical Wiener process.
By a simple application of Itô’s formula, the law (νt(dx)) of Ztprovides a solution to (4).
According to Exercise 7.3.3 of [47] (Krylov estimates) it is possible to show the existence of
(t,x) ?→ p(t,x) in L2[[0,1]×R) being density of (νt(dx)). In particular for almost all t ∈ [0,1],
νt(dx) admits a density.
1. Assumption 3.15 is verified for σ(t,x) ≡ σ, being σ a positive real constant
Proposition 3.17. Let (Xt,0 ≤ t ≤ 1) be a continuous finite quadratic variation process with
X0= 0, and d[X]t= (σ(t,Xt))2dt, where σ fulfills Assumption 3.15. Suppose that A = A1
the following statements are true.
X. Then
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1. X is an A-martingale if and only if, for every 0 ≤ t ≤ 1, Xt
of equation dZ = σ(·,Z)dB,Z0= 0. In particular, if σ ≡ 1, X is a weak Brownian motion
of order 1, if and only if it is an A1
2. Suppose that d[X]t= ftdt, with f being B([0,1])-measurable and bounded. If X is a weak
Brownian motion of order k = 1, then X is an A-semimartingale. Moreover the process
?·
is an A-martingale.
law
= Zt, for every (Z,B) solution
X-martingale.
X +
0
(1 − fs)Xs
2s
ds.
Proof.
1. Using Itô’s formula recalled in Proposition 2.10 we can write, for every 0 ≤ t ≤ 1 and
ψ = ∂xΨ according to the definition of A1
?t
?
X
0
ψ(s,Xs)d−Xs
= Ψ(t,Xt) − Ψ(0,X0)
(5)
−
?t
0
∂sΨ +1
2∂(2)
xxΨσ2
?
(s,Xs)ds.
For every 0 ≤ t ≤ 1, we denote with µt(dx) the law of Xt. If X is an A1
(5) we derive
X-martingale, from
0=
?
1
2
R
Ψ(t,x)µt(dx) −
?t
?
R
Ψ(0,x)µ0(dx) −
?t
0
?
R
∂sΨ(s,x)µs(dx)ds
−
0
?
R
∂(2)
xxΨ(s,x)σ(s,x)2µs(dx)ds.
(6)
In particular, the law of X solves equation (4).
On the other hand, let (Z,B) be a solution of equation Z =
Z fulfills equation (6) too. Indeed, Z is a finite quadratic variation process with d[Z]t=
(σ(t,Zt))2dt which is an A1
must have the same law as Zt. This establishes the direct implication of point 1.
Suppose, on the contrary, that Xthas the same law as Zt, for every 0 ≤ t ≤ 1. Using the
fact that Z is an A1
?
?·
0σ(s,Zs)dBs. The law of
X-martingale by point 2. of Remark 3.3. By Assumption 3.15 Xt
X-martingale which solves equation (5) we get
?t
E
Ψ(t,Zt) − Ψ(0,Z0) −
0
?
∂sΨ +1
2∂(2)
xxΨσ2
?
(s,Zs)ds
?
= 0,
for every Ψ in C1,2([0,1]×R) with ∂xΨ = ψ according to A1
Zt, for every 0 ≤ t ≤ 1, equality (5) implies that
??·
The proof of the first point is now achieved.
X. Since Xthas the same law as
E
0
ψ(t,Xt)d−Xt
?
= E
??·
0
ψ(t,Zt)d−Zt
?
= 0,
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2. Suppose that σ(t,x)2= ft, for every (t,x) in [0,1] × R. Let Ψ be in C1,2([0,1] × R) such
that ψ(·,X) = ∂xΨ(·,X) belongs to A1
?t
with
YΨ
t = Ψ(t,Xt) − Ψ(0,X0) −
0
Moreover X is a weak Brownian motion of order 1. This implies E?YΨ
??t
Since the law of Xtis N(0,t), by Fubini’s theorem and integration by parts on the real line
we obtain
??t
This concludes the proof of the second point.
X. Proposition 2.10 yields
?t
?t
0
ψ(s,Xs)d−Xs= YΨ
t +1
2
0
∂(2)
xxΨ(s,Xs)(1 − fs)ds,
0 ≤ t ≤ 1,
∂sΨs(s,Xs)ds −1
2
?t
0
∂(2)
xxΨ(s,Xs)ds.
t
?
= 0, for every
0 ≤ t ≤ 1. We derive that
E
0
ψ(s,Xs)d−Xs+1
2
?t
0
∂(2)
xxΨ(s,Xs)(fs− 1)ds
?
= E?YΨ
t
?= 0.
E
0
∂(2)
xxΨ(s,Xs)(fs− 1)ds
?
= E
??t
0
ψ(s,Xs)(1 − fs)Xs
s
ds
?
.
Remark 3.18. In the statement of Proposition 3.17, we may not suppose a priori the uniqueness
for PDE (4). We can replace it with the following.
Assumption 3.19. – σ is non-degenerate.
– Let µt(dx) be the law of Xt,t ∈ [0,1]. We suppose that the Borel finite measure µt(dx)dt on
[0,1] × R admits a density (t,x) ?→ q(t,x) in L2([0,1] × R).
In fact, the same proof as for item 1. works, taking into account item 2. of Remark 3.16 the
difference p−q belongs to L2([0,1]×R); by Theorem 3.8 of [8] p = q and so the law of Xtand Zt
are the same for any t ∈ [0,1].
From [20] we can extract an example of an A-semimartingale which is not a semimartingale.
Example 3.20. Suppose that (Bt,0 ≤ t ≤ 1) is a Brownian motion on the probability space (Ω,G,P),
being G some filtration on (Ω,F,P). Set
?Bt,
Then X is a continuous weak Brownian motion of order 1, which is not a G-semimartingale.
Moreover it is possible to show that d[X]t= ftdt, with f = I[0,1
thanks to point 2. of previous Proposition 3.17, X +?·
A natural question is the following. Supposing that X is an A-martingale with respect to a proba-
bility measure Q equivalent to P, what can we say about the nature of X under P? The following
Proposition provides a partial answer to this problem when A = A1
Xt=
0 ≤ t ≤1
1
2< t ≤ 1.
2
B 1
2+ (√2 − 1)Bt−1
2,
2]+ (√2 − 1)2I[1
(1−fs)Xs
2s
ds is an A1
2,1]. In particular,
X-martingale. In fact
0
the notion of quadratic variation is not affected by the enlargement of filtration.
X.
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Proposition 3.21. Let X be as in Proposition 3.17, and σ satisfy Assumption 3.15. Assume,
furthermore, that X is an A1
that the solution (νt(dx)) of (4) admits a density for every t ∈ (0,1]. Then the law of Xt is
absolutely continuous with respect to Lebesgue measure, for all t ∈ (0,1].
X-martingale under a probability measure Q with P << Q. Suppose
Proof. Since P << Q, for every 0 ≤ t ≤ 1, the law of Xtunder P is absolutely continuous with
respect to the law of Xtunder Q. Then it is sufficient to observe that by Proposition 3.17, for all
0 ≤ t ≤ 1, the law of Xtunder Q is absolutely continuous with respect to Lebesgue. By Proposition
3.17, the law of Xt is equivalent to the law νt of Zt for every t ∈ [0,1]. The conclusion follows
because νtis absolutely continuous.
Corollary 3.22. Let X be as in Proposition 3.17, and σ satisfy Assumption 3.15. Assume, fur-
thermore, that X is an AX-martingale under a probability measure Q with P << Q, Then the law
of Xtis absolutely continuous with respect to Lebesgue measure, for every 0 ≤ t ≤ 1.
Proof. Clearly A1
3.21.
Xis contained in AX. The result is then a consequence of previous Proposition
Proposition 3.23. Let (Xt,0 ≤ t ≤ 1) be a continuous weak Brownian motion of order 8. Then,
for every ψ : [0,1] × R → R, Borel measurable with polynomial growth, the forward integral
?·
E
0
0ψ(t,Xt)d−Xt, exists and
??·
ψ(t,Xt)d−Xt,
?
= 0.
In particular, X is an AX-martingale.
Proof. Let ψ : [0,1] × R → R be Borel measurable and t in 0 ≤ t ≤ 1 be fixed. Set
IX
ε(t) = I(ε,ψ(·,X),X)
being B a Brownian motion on a filtered probability space (ΩB,FB,PB).
IB
ε(t) = I(ε,ψ(·,B),B),
Since X is a weak Brownian motion of order 8, it follows that
???IX
We show now that IB
E
ε(t) − IX
δ(t)??4?
= EPB???IB
ε(t) − IB
δ(t)??4?
,
∀ ε,δ > 0.
ε(t) converges in L4(Ω). This implies that IX
In [42], chapter 3.5, it is proved that IB
limit equals the Itô integral?t
estimate, for every p > 4 :
EPB???IB
for some positive constant c. This implies the uniformly integrability of the family of random
variables?(IB
Consequently,
E[I(t)] = 0, being I(t) the limit in L2(Ω) of random variables having zero expectation.
ε(t) is of Cauchy in L4(Ω).
ε(t) converges in probability when ε goes to zero, and the
0ψ(s,Bs)dBs. Applying Fubini’s theorem for Itô integrals, theorem
45 of [34], chapter IV and Burkholder-Davies-Gundy inequality, we can perform the following
ε(t)??p?
≤ c sup
t∈[0,1]EPB[|ψ(t,Bt)|p] < +∞,
ε(t))4?
ε>0and therefore the convergence in L4(ΩB,PB) of?IB
ε(t)?
ε>0.
?IX
ε(t)?
ε>0converges in L4(Ω) toward a random variable I(t). It is clear that
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To conclude we show that Kolmogorovlemma applies to find a continuous version of (I(t),0 ≤ t ≤ 1).
Let 0 ≤ s ≤ t ≤ 1. Applying the same arguments used above
?
E
|I(t) − I(s)|4?
≤ sup
u∈[0,1]
EPB?
|ψ(u,Bu)|4?
|t − s|2, c > 0.
Remark 3.24. If X is a 4-order weak Brownian motion than, using the techniques of proof of
previous result, that W has quadratic variation [X]t= t.
3.3. Optimization problems and A-martingale property
3.3.1. Gâteaux-derivative: recalls
In this part of the paper we recall the notion of Gâteaux differentiability and we list some related
properties.
Definition 3.25. A function f : A → R is said Gâteaux-differentiable at π ∈ A, if there exists
Dπf : A → R such that
f(π + εθ) − f(π)
ε
lim
ε→0
= Dπf(θ),
∀θ ∈ A.
If f is Gâteaux-differentiable at every π ∈ A, then f is said Gâteaux-differentiable on A.
Definition 3.26. Let f : A → R. A process π is said optimal for f in A if
f(π) ≥ f(θ),
∀θ ∈ A.
We state this useful lemma omitting its straightforward proof.
Lemma 3.27. Let f : A → R. For every π and θ in A define fπ,θ: R −→ R in the following way:
fπ,θ(λ) = f(π + λ(θ − π)).
Then it holds:
1. f is Gâteaux-differentiable if and only if for every π and θ in A, fπ,θis differentiable on R.
Moreover f′
π,θ(λ) = Dπ+λ(θ−π)f(θ − π).
2. f is concave if and only if fπ,θis concave for every π and θ in A.
Proposition 3.28. Let f : A → R be Gâteaux-differentiable. Then, if π is optimal for f in A,then
Dπf = 0. If f is concave
π is optimal for f in A ⇐⇒
Dπf = 0.
Proof. It is immediate to prove that π is optimal for f if and only if λ = 0 is a maximum for
fπ,θ, for every θ in A. By Lemma 3.27 f
easily.
′
π,θ(0) = Dπf(θ), for every θ in A. The conclusion follows
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3.3.2. An optimization problem
In this part of the paper F will be supposed to be a measurable function on (Ω × R,F ⊗ B(R)),
almost surely in C1(R), strictly increasing, with F′being the derivative of F with respect to x,
bounded on R, uniformly in Ω. In the sequel ξ will be a continuous finite quadratic variation process
with ξ0= 0.
The starting point of our construction is the following hypothesis.
Assumption 3.29.
2. Every θ in A is ξ-improperly forward integrable, and
1. If θ belongs to A, then θI[0,t]belongs to A for every 0 ≤ t < 1.
E
?????
?1
0
θtd−ξt
????+
????
?1
0
θ2
td[ξ]t
????
?
< +∞.
Definition 3.30. Let θ be in A. We denote
Lθ=
?1
0
θtd−ξt−1
2
?1
0
θ2
td[ξ]t, dQθ=
F′(Lθ)
E[F′(Lθ)]
and we set f(θ) = E?F(Lθ)?.
We observe that point 2. of Assumption 3.29 and the boundedness of F′imply that E???F(Lθ)???<
Remark 3.31. Point 2. of Assumption 3.29 implies that E[|ξt| + [ξ]t] < +∞, for every 0 ≤ t ≤ 1.
This is due to the fact that A must contain real constants.
We are interested in describing a link between the existence of an optimal process for f in A and
the A-semimartingale property for ξ under some probability measure equivalent to P, depending
on the optimal process.
+∞. Therefore f is well defined.
Lemma 3.32. The function f is Gâteaux-differentiable on A. Moreover for every π and θ in A
?
If F is concave, then f inherits the property.
Dπf(θ) = E
F′(Lπ)
?1
0
θtd−
?
ξt−
?t
0
πsd[ξ]s
??
.
Proof. Regarding the concavity of f, we recall that if F is increasing and concave, it is sufficient
to verify that, for every θ and π in A, it holds
Lπ+λ(θ−π)− Lπ− λ?Lθ− Lπ?≥ 0,
A short calculation shows that, for every 0 ≤ λ ≤ 1,
Lπ+λ(θ−π)− Lπ− λ?Lθ− Lπ?=1
Using the differentiability of F we can write
?
0 ≤ λ ≤ 1.
2λ(1 − λ)
?1
0
(θt− πt)2d[ξ]t≥ 0.
aε=1
ε(f(π + εθ) − f(π)) = E
Hε
π,θ
?1
0
F′?Lπ+ µεHε
π,θ
?dµ
?
,
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with
Hε
π,θ=
?1
0
θtd−ξt−1
2
?1
0
(θ2
tε + 2θtπt)d[ξ]t.
The conclusion follows by Lebesgue dominated convergence theorem, which applies thanks to the
boundedness of F′and point 2. in Assumption 3.29.
Putting together Lemma 3.32 and Proposition 3.28 we can formulate the following.
Proposition 3.33. If a process π in A is optimal for θ ?→ E?F?Lθ??, then the process ξ−?·
Proof. Thanks to Lemma 3.32 and point 1. in Assumption 3.29, for every θ in A and 0 ≤ t ≤ 1
?
EQπ??t
0πtd[ξ]t
is an A-martingale under Qπ. If F is concave the converse holds.
0=
Dπf(θI[0,t]) = E
F′(Lπ)
?t
0
θsd−
?
ξs−
?s
0
πrd[ξ]r
??
=
0
θsd−
?
ξs−
?s
0
πrd[ξ]r
??
.
The following Proposition describes some sufficient conditions to recover the semimartingale prop-
erty for ξ with respect to a filtration G on (Ω,F), when the set A is made up of G-adapted
processes. It can be proved using Proposition 3.7.
Proposition 3.34. Assume that ξ is adapted with respect to some filtration G and that A satisfies
the hypothesis D with respect to G. If a process π in A is optimal for θ ?→ E?F(Lθ)?, then the
If F is concave, then the converse holds.
process ξ −?·
0βtd[ξ]tis a G-martingale under P, where β = π +
1
pπ
d[pπ,ξ]
d[ξ,ξ], and pπ= E
?
dP
dQπ| G·
?
.
Proof. Thanks to point 2. of Assumption 3.29, for every 0 ≤ t < 1, the random variable ξt−
?t
35, chapter III, of [34], we get the necessity condition. As far as the converse is concerned, we
observe that, thanks to the hypotheses on A, if ξ−?·
is a G-martingale starting at zero with zero expectation.
This concludes the proof.
0πsd[ξ]sis in L1(Ω) and so in L1(Ω,Qπ) being
state that ξ −?·
A, the process?·
Proposition 3.35. Suppose that there exists a measurable process (γt,0 ≤ t ≤ 1) such that the
process ξ −?·
2. Assume, furthermore, the existence of a sequence of processes (θn)n∈N⊂ A with
??1
If there exists an optimal process π, then d[ξ]{t ∈ [0,1),γt?= πt} = 0, almost surely.
dQπ
dP
bounded. Then Proposition 3.7 applies to
0πtd[ξ]tis a G-martingale under Qπ. Using Meyer Girsanov theorem, i.e. Theorem
0πtd[ξ]tis a G-martingale, then for every θ in
0θtd−?
ξt−?t
0πsd[ξ]s
?
0γtd[ξ]tis an A-martingale.
1. If γ belongs to A then γ is optimal for θ ?→ E?Lθ?.
lim
n→+∞E
0
|θn
t− γt|2d[ξ]t
?
= 0.
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19
Proof.
1. The identity function F(ω,x) = x is of course strictly increasing and concave. The
first point is an obvious consequence of Proposition 3.33.
2. Again by Proposition 3.33 and additivity, we deduce that a process π is optimal θ ?→ E(Lθ)
if and only if the process
optimal if and only if for every θ is in A it holds: E
words π is optimal if and only if γ − π belongs to the orthogonal of A with respect to the
Hilbert space H of measurable processes R : [0,T]×Ω → R equipped with the inner product
?θ,ℓ? = E
belongs to the closure of A onto H. Finally γ − π has to vanish.
?·
0(γt− πt)d[ξ]tis an A-martingale under P. Consequently π is
??1
0θt(γt− πt)d[ξ]t
?
= 0. In other
??1
0θsℓsd[ξ]s
?
. By the assumption of item 2. it follows that γ and therefore γ − π
4. The market model
We consider a market offering two investing possibilities in the time interval [0,1]. Prices of the two
traded assets follow the evolution of two stochastic processes?S0
S0
t= (exp(Vt),0 ≤ t ≤ 1),
where (Vt,0 ≤ t ≤ 1) is a positive process starting at zero with bounded variation, and S is a
continuous strictly positive process, with finite quadratic variation.
1. If V =?·
is a riskless asset, being that assumption not necessary to develop our calculus. We only need
to suppose that S0is less risky then S.
t,0 ≤ t ≤ 1?and (St,0 ≤ t ≤ 1).
We could assume that
Remark 4.1.
price process of the so called money market account. Here we do not need to assume that V
0rsds, being (rt,0 ≤ t ≤ 1) the short interest rate, S0represents the
2. Assuming that S has a finite quadratic variation is not restrictive at least for two reasons.
Consider a market model involving an inside trader: that means an investor having additional
informations with respect to the honest agent. Let F and G be the filtrations representing
the information flow of the honest and the inside investor, respectively. Then it could be
worthwhile to demand the absence of free lunches with vanishing risk (FLVR) among all
simple F-predictable strategies. Under the hypothesis of absence of (FLVR), by theorem 7.2,
page 504 of [11], S is a semimartingale on the underlying probability space (Ω,P,F). On the
other hand S could fail to be a G-semimartingale, since (FLVR) possibly exist for the insider.
Nevertheless, the inside investor is still allowed to suppose that S has finite quadratic variation
thanks to Proposition 2.7.
Secondly, as already specified in the introduction, if we want to include S as a self-financing-
portfolio, we have to require that
finite quadratic variation, see Proposition 4.1 of [41].
?·
0Sd−S exists. This is equivalent to assume that S has
4.1. Portfolio strategies
We assume the point of view of an investor whose flow of information is modeled by a filtration
G= (Gt)t∈[0,1]of F, which satisfies the usual assumptions.
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20
We denote with C−
on each compact set of [0,1).
Definition 4.2. A portfolio strategy is a couple of G-adapted processes φ =??h0
b([0,1)) the set of processes which have paths being left continuous and bounded
t,ht
?,0 ≤ t < 1?.
The market value X of the portfolio strategy φ is the so called wealth process X = h0S0+ hS.
We stress that there is no point in defining the portfolio strategy at the end of the trading period,
that is for t = 1. Indeed, at time 1, the agent has to liquidate his portfolio.
?h0,h?
?·
Definition 4.3. A portfolio strategy φ =
C−
b([0,1)), the process h is locally S-forward integrable and its wealth process X verifies
is self-financing if both h0and h belong to
X = X0+
0
h0
tdS0
t+
?·
0
htd−St.
(7)
Remark 4.4. When S is a G-semimartingale, if h ∈ C−
previous forward integral coincide with classical Itô integrals, see Proposition 2.7.
b([0,1)) is locally S-forward integrable and
The interpretation of the first two items in definition 4.3 is straightforward: h0and h represent,
respectively, the number of shares of S0and S held in the portfolio; X is its market value. The self-
financing condition (7) seems to be an appropriate formalization of the intuitive idea of trading
strategy not involving exogenous sources of money. Among its justifications we can include the
following ones.
As already explained in the introduction, the discrete time version of condition (7) reads as the
classical self-financing condition. Furthermore, if S is a G-semimartingale, forward integrals of G-
adapted processes with left continuous and bounded paths, agree with classical Itô integrals, see
Proposition 2.8 and 2.7.
It is natural to choose as as numéraire the positive process S0. That means that prices will be
expressed in terms of S0. We could denote with?Y the value of a stochastic process (Yt,0 ≤ t ≤ 1)
The following lemma shows that, as well as in a semimartingale model, a portfolio strategy which
is self-financing is uniquely determined by its initial value and the process representing the number
of shares of S held in the portfolio. We remark that previous definitions and considerations can be
made without supposing that the investor is able to observe prices of S and S0. However, we need
to make this hypothesis for the following characterization of self-financing portfolio strategies.
discounted with respect to S0:?Yt= Yt(S0
t)−1, for every 0 ≤ t ≤ 1.
Assumption 4.5. From now on we suppose that S and S0are G-adapted processes.
Remark 4.6. Indeed, for simplicity of the formulation, we will suppose in most of the proofs in
the sequel that V ≡ 0 so that S0≡ 1. Usual rules of calculus via regularization allow to prove
statements to the case of general S0. In that case the role of the wealth process (resp. the stock
price) X (resp. S) will be replaced by˜ X (resp.˜S). With our simplifying convention we will wave
X =˜ X, S =˜S.
Proposition 4.7. Let (ht,0 ≤ t < 1) be a G-adapted process in C−
forward integrable, and X0be a G0-random variable. Suppose V ≡ 0. Then the couple
φ =?h0
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b([0,1)), which is locally S-
t,ht,0 ≤ t < 1?,
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21
where h0
t= Xt− htSt, X defined as
X = X0+
?·
0
htd−St,
(8)
is a self-financing portfolio strategy with wealth process X.
Proof. Let h, X0and X be as in the second part of the statement. It is clear that
h0= ((Xt− htSt),0 ≤ t < 1) is G-adapted and belongs to C−
process corresponding to the strategy φ = (h0,h) is equal to X. The conclusion follows by (7).
b([0,1)). By construction, the wealth
Proposition 4.7 leads to conceive the following definition.
Definition 4.8.
variable X0, and a process h in C−
2. In the sequel we let us employ the term portfolio to denote the process h (in a self-financing
portfolio), representing the number of shares of S held. Without further specifications the
initial wealth of an investor will be assumed to be equal to zero.
1. A self-financing portfolio is a couple (X0,h) of a G0-measurable random
b([0,1)) which is G-adapted and locally S-forward integrable.
Some conditions to insure the existence of chain-rule formulae, when the semimartingale property
of the integrator process fails to hold, can be found in [18].
Assumption 4.9. We assume the existence of a real linear space of portfolios A, that is of G-
adapted processes h belonging to C−
b([0,1)), which are locally S-forward integrable. The set A will
represent the set of all admissible strategies for the investor.
We proceed furnishing examples of sets behaving as the set A in Assumption 4.9.
4.2. About some classes of admissible strategies
The aim of this section is to provide some classes of mathematically rigorous admissible strate-
gies. We will leave most of technical justifications to the reader; they are based on calculus via
regularization, see [42] for a recent survey.
4.2.1. Admissible strategies via Itô fields
Adapting arguments developed in [18], we consider the following framework. Given a G-adapted
process (ξt) we denote by C1
H(t,x),0 ≤ t ≤ 1,x ∈ R is a random field of the form
n
?
where f : Ω × R → R belongs to C1(R) almost surely and it is G0-measurable for every x, H and
ai: [0,1]×R×Ω → R, i = 1,...,n are G-adapted for every x, almost surely continuous with their
partial derivatives with respect to x in (t,x) and it holds
?t
ξ(G) the class of processes of the form H(t,ξt),0 ≤ t ≤ 1) where
H(t,x) = f(x) +
i=1
?t
0
ai(s,x)dNi
s,
0 ≤ t ≤ 1,
(9)
∂xH(t,x) = ∂xf(x) +
n
?
i=1
0
∂xai(s,x)dNi
s,
0 ≤ t ≤ 1.
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The following Proposition can be proved using the machinery developed in [18]
Proposition 4.10. Let A be the set of processes (ht,0 ≤ t < 1) such that for every 0 ≤ t < 1
the process in hI[0,t]belongs to C1
Assumption 4.9.
S(G). Then A is a real linear space satisfying the hypotheses of
4.2.2. Admissible strategies via Malliavin calculus
Malliavin calculus represents a very efficient way to introduce a class of admissible strategies if the
logarithm of the underlying price is a Gaussian non-semimartingale or if anticipative strategies are
admitted. Basic notations and definitions concerning Malliavin calculus can be found for instance
in [32] and [31].
We suppose that (Ω,F,F,P) is the canonical probability space, meaning that Ω = C ([0,1],R), P
is the Wiener measure, W is the Wiener process, F is the filtration generated by W and the P-null
sets and F is the completion of the Borel σ-algebra with respect to P.
For p > 1,k ∈ N∗, Dk,pwill denote the classical Wiener-Sobolev spaces.
For any p ≥ 2, L1,pdenotes the space of all functions u in Lp(Ω × [0,1]) such that ut belongs
to D1,pfor every 0 ≤ t ≤ 1 and there exists a measurable version of (Dsut,0 ≤ s,t ≤ 1) with
?1
We recall that D1,2is dense in L2(Ω), L1,2⊂ Domδ, and that if u belongs to L1,2then, for
each 0 ≤ t ≤ 1, uI[0,t]is still in L1,2. In particular it is Skorohod integrable. We will use the
notation δ?uI[0,t]
Definition 4.11. For every p ≥ 2, L1,p
that limε→0Dtut−εexists in Lp(Ω×[0,1]). The limiting process will be denoted by?D−
Techniques similar to those of [31, 32] allow to prove the following.
?u1,...,un?, n > 1, be a vector of left continuous processes with
that the random variable |vt|+sups∈[0,1]|Dsvt| is bounded. Then for every ψ in C1(Rn) ψ(u)v and
v are forward integrable with respect to W. Furthermore ψ(u) is forward integrable with respect to
?·
ψ(ut)d−
0
0E
its domain is denoted by Domδ. An element u belonging to Domδ is said Skorohod integrable.
?
||Dut||p
L2([0,1])
?
dt < ∞. The Skorohod integral δ is the adjoint of the derivative operator D;
?
=
?t
0usδWs, for each u in L1,2. The process
??t
0usδWs,0 ≤ t ≤ 1
?
is mean
square continuous and then it admits a continuous version, which will be still denoted by?·
0utδWt.
− will be the space of all processes u belonging to L1,psuch
tut,0 ≤ t ≤ 1?.
Proposition 4.12. Let u =
bounded paths and in L1,p
−, with p > 4. Let v be a process in L1,2
− with left continuous paths such
0vtd−Wtand
?·
0
??t
vsd−Ws
?
=
?·
0
ψ(ut)vtd−Wt.
Regarding the price of S we make the following assumption.
Assumption 4.13. We suppose that S = S0exp??·
L1,2
?
0σtdWt+?·
0
?µt−1
2σ2
t
?dt?, where µ and σ are
F-adapted, µ belongs to L1,qfor some q > 4, σ has bounded and left continuous paths, it belongs
−∩ L2,2and the random variable
sup
t∈[0,1]
|σt| + sup
s∈[0,1]
|Dsσt|sup
s,u∈[0,1]
|DsDuσt|
?
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is bounded.
Remark 4.14. By Remark of page 32, section 1.2 of [31] σ is in L1,2
− and D−σ = 0.
Performing usual technicalities as in [31, 32] it is possible to prove that the process log(S) belongs
to L1,q
−.
Proposition 4.15. Let A be the set of all G-adapted processes h in C−
0 ≤ t < 1, the process hI[0,t]belongs to L1,p
the hypotheses of Assumption 4.9.
b([0,1)), such that for every
−, for some p > 4. Then A is a real linear space satisfying
Proof. Let h be in A. We set A = log(S)−log(S0)+1
thanks to Proposition 2.11, for every 0 ≤ t < 1, hI[0,t]is S-forward integrable if and only if hI[0,t]S
is forward integrable with respect to A. Let 0 ≤ t < 1, be fixed. Each component of the vector
process u =?hI[0,t],log(S)?belongs to L1,p
to?·
2
?·
0σ2
tdt =?·
0σtdWt+?·
0µtdt. We recall that,
− for some p > 4 and it has left continuous and bounded
paths. We can thus apply Proposition 4.12 to state that hI[0,t]S is forward integrable with respect
0σtdWt. This implies that hI[0,t]S is A-forward integrable. Letting t vary in [0,1) we find that
h is S-improperly integrable and we conclude the proof.
4.2.3. Admissible strategies via substitution
Let F = (Ft)t∈[0,1]be a filtration on (Ω,F,P), with F1 = F, and G an F measurable random
variable with values in Rd. We set Gt= (Ft∨ σ(G)), and we suppose that G is right continuous:
Gt=
ε>0
?
(Ft+ε∨ σ(G)).
In this section PF(PG, resp.) will denote the σ-algebra of F (of G, resp.)-predictable processes.
E will be the Banach space of all continuous functions on [0,1] equipped with the uniform norm
||f||E= supt∈[0,1]|f(t)|.
Definition 4.16. An increasing sequence of random times (Tk)k∈Nis said suitable if
P?∪+∞
Let Ap,γ(G) be the set of processes (ut) where ut= h(t,G) where h(t,x) is a random field fulfilling
the following Kolmogorov type conditions: there is a suitable sequence of stopping times (Tk) for
which
?
t∈[0,Tk]
k=0{Tk= 1}?= 1.
E
sup|h(t,x) − h(t,y)|p
?
≤ c|x − y|γ,
∀x,y ∈ C.
We assume that S and S0are F-adapted, and that S is an F-semimartingale.
We observe that this situation arises when the investor trades as an insider, that is having an
extra information about prices, at time 0, represented by the random variable G.
Performing substitution formulae as in [38, 41, 40, 17], it is possible to establish the following
result.
Proposition 4.17. Let A be the set of processes h such that, for every 0 ≤ t < 1, the process
hI[0,t]belongs to Ap,γfor some p > 1 and γ > 0. Then A satisfies the hypotheses of Assumption
4.9.
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4.3. Completeness and arbitrage: A-martingale measures
Definition 4.18. Let h be a self financing portfolio in A which is S-improperly forward integrable
and X is its wealth process. Then h is an arbitrage if X1 = limt→1Xt exists almost surely,
P({X1≥ 0}) = 1 and P({X1> 0}) > 0.
Definition 4.19. We say that the market is A-arbitrage free if no self-financing strategy h in
A is an arbitrage.
Definition 4.20. A probability measure Q ∼ P is said A-martingale measure if under Q the
process S is an A-martingale according to definition 3.1.
For the following Proposition the reader should keep in mind the notation in equality (2). We omit
its proof which is a direct application of Corollary 3.22.
Proposition 4.21. Let A = AS. Suppose that d[S]t= σ(t,St)2S2
3.15. Moreover we suppose that the unique solution of equation (4) admits a density for 0 < t ≤ 1.
If there exists a A-martingale measure then the law of Stis absolutely continuous with respect to
Lebesgue measure, for every 0 < t ≤ 1.
Proposition 4.22. If there exists an A-martingale measure Q, the market is A-arbitrage free.
Proof. Suppose again that V ≡ 0 and that h is an A-arbitrage. Since S is an A-martingale under
Q, we find EQ[X1] = EQ[?1
We proceed discussing completeness.
tdt, where σ satisfies Assumption
0htd−St] = 0. This contradicts the arbitrage condition Q({X1> 0}) >
0.
Definition 4.23. A contingent claim C is an F-measurable random variable. We denote˜C =
C
S0
the investor is interested in.
T. L will be a set of F-measurable random variables; it will represent all the contingent claims
Definition 4.24.
portfolio (X0,h) with h in A, which is S-improperly forward integrable, such that the corre-
sponding wealth process X verifies limt→1Xt= C, almost surely. The portfolio h is said the
replicating or hedging portfolio for C, X0is said the replication price for C.
2. The market is said to be (A,L)-attainable if every contingent claim in L is attainable trough
a portfolio in A.
Assumption 4.25. For every G0-measurable random variable η, and (ht) in A the process u = hη,
belongs to A.
Proposition 4.26. Suppose that the market is A-arbitrage free, and that Assumption 4.25 is
realized. Then the replication price of an attainable contingent claim is unique.
1. A contingent claim C is said A-attainable if there exists a self financing
Proof. Let (X0,h) and (Y0,k) be two replicating portfolios for a contingent claim C, with h and k
in A, and wealth processes X and Y , respectively. We have to prove that
P ({X0− Y0?= 0}) = 0.
Suppose, for instance, that P (X0− Y0> 0) ?= 0. We set A = {X0− Y0> 0}. By Assumption
4.25, IA(k −h) is a portfolio in A with wealth process IA(Yt−Xt). Since both (X0,h) and (Y0,k)
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25
replicate C, limt→1IA(Yt− Xt) = IA(X0− Y0), with P({IA(X0− Y0> 0)}) > 0. Then IA(k − h)
is an A-arbitrage and this contradicts the hypothesis.
Proposition 4.27. Suppose that there exists an A-martingale measure Q. Then the following
statements are true.
1. Under Assumption 4.25, the replication price of an A-attainable contingent claim C is unique
and equal to EQ??C | G0
every A-attainable contingent claim C. In particular, if the market is (A,L)-attainable and L
is an algebra, all A-martingale measures coincide on the σ-algebra generated by all bounded
discounted contingent claims in L.
Proof. Suppose againV ≡ 0. Let (X0,h) be a replicating A-portfolio for C. Then
?
.
2. Let G0 be trivial. If Q and Q1 are two A-martingale measures, then EQ[?C] = EQ1[?C], for
EQ[C | G0] = X0+ EQ
?
??1
0
htd−St| G0
?
.
We observe that EQ??1
0. This implies point 1.
0htd−St| G0
= 0. In fact, if η is a G0-measurable random variable, then,
0htd−St
thanks to Assumption 4.25, ηh belongs to A, so as to have EQ???1
If G0 is trivial, we deduce that, if Q and Q1 are two A-martingale measures, EQ[C] = EQ1[C],
for every A-attainable contingent claim. The last point is a consequence of the monotone class
theorem, see theorem 8, chapter 1 of [34].
?
η
?
= EQ??1
0ηhtd−St
?
=
4.4. Hedging
In this part of the paper we price contingent claims via partial differential equations. In particular,
within a non-semimartingale model, we emphasize robustness of Black-Scholes formula for Euro-
pean, Asian and some path dependent contingent claims depending on a finite number of dates of
the underlying price.
We suppose here that d[S]t= σ2(t,St)S2
We suppose the existence of constants c1,c2such that 0 < c1≤ σ ≤ c2.
Similar results were obtained by [45] and [49]. Examples of non-semimartingale processes S of that
type can be easily constructed. They are related to processes X such that [X] = const t. A typical
example is a Dirichlet process which can be written as Brownian motion plus a zero quadratic
variation term. A not so well-known example is given by bifractional Brownian motion X = BH,K
for indices H ∈]0,1[,K ∈]0,1] such that HK =1
semimartingale nor a Dirichlet process.
tdt and dVt= rdt, with r > 0 and σ : [0,1]×(0,+∞) → R.
2, see for instance [37]. This process is neither a
Proposition 4.28. Let ψ be a function in C0(R). Suppose that there exists (v(t,x),0 ≤ t ≤ 1,x ∈ R)
of class C1,2([0,1) × R) ∩ C0([0,1] × R), which is a solution of the following Cauchy problem
?
v(1,y)
∂tv(t,y) +1
2(? σ(t,y))2y2∂(2)
yyv(t,y)=
=
0
?ψ(y),
on [0,1) × R
(10)
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where
?? σ(t,y) = σ(t,yert)∀(t,y) ∈ [0,1] × R,
∀y ∈ R.
?ψ(y) = ψ(yer)e−r
ht= ∂yv(t,?St),
Set
0 ≤ t < 1,X0= v(0,S0).
Then (X0,h) is a self-financing portfolio replicating the contingent claim ψ(S1).
Proof. Again, for simplicity, we consider the case r = 0. Assumption 4.5 tells us that h is a G-
adapted process in C−
Proposition 2.10, recalling equation (10), equalities (7) we find that
b([0,1)). By Proposition 2.10, h is locally S-forward integrable. Applying
Xt= v(t,St),
∀0 ≤ t < 1.
In particular X0+ limt→1
?t
0hsd−Ssexists finite and coincides with v(1,S1) = ψ(S1).
Remark 4.29. In particular, under some minimal regularity assumptions on σ and no degeneracy,
the market is (AS,L)-attainable, if L equals the set of all contingent claims of type ψ(S1) with ψ
in C0(R) with linear growth.
Enlarging suitably A and solving successively and recursively equations of the type (10), it is
possible to replicate contingent claims of the type C = ψ(Xt1,··· ,Xtn) with 0 ≤ t1< ··· < tn= 1
and ψ : Rn→ R continuous with polynomial growth.
The proposition below provides a suitable framework for this.
Proposition 4.30. Let r = 0 so V ≡ 0. Suppose d[S]t= σ2(t,St)S2
with polynomial growth. Let 0 = t0< t1< ... < tn= 1, n ≥ 2. Suppose that there exist functions
v1,...,vnsuch that
– vi∈ C1,2([ti−1,ti) × Ri) ∩ C0([ti−1,ti] × Ri]), 1 ≤ i ≤ n;
– and denoting shortly vi(t,y) := vi(t,y1,...,yi−1,y) for 1 ≤ i ≤ n we have
?
tdt and ψ a function in C0(Rn)
∂tvn(t,y) +1
vn(1,y1,...,yn−1,y) = ψ(y1,...,yn−1,y)
2σ2(t,y)y2∂(2)
yyvn(t,y) = 0
on [tn−1,1) × R
(11)
and for i = 1,...,n − 1
?
∂tvi(t,y) +1
vi(ti,y1,...,yi−1,y) = vi+1(ti,y1,...,yi−1,y,y).
2σ2(t,y)y2∂(2)
yyvi(t,y) = 0
on [ti−1,ti) × R
(12)
In particular v1(t1,y) = v2(t1,y,y).
Setting
ht
=
I[0,t1](t)∂yv1(t,St) +
n
?
i=2
I(ti−1,ti](t)∂yvi(t,St1,...,Sti−1,St)
X0
=
v1(0,S0) .
Then (X0,h) is a self-financing portfolio replicating the contingent claim ψ(St1,...,Stn).
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The result of Proposition 4.28 can also be adapted to hedge Asian contingent claims, that is
contingent claims C depending on the mean of S over the traded period: C = ψ
for some ψ in C0(R).
?
1
S1
??1
0Stdt
??
S1,
Proposition 4.31. Suppose that σ(t,x) = σ, for every (t,x) in [0,1] × R, for some σ > 0. Let ψ
be a function in C0(R) and v(t,y) a continuous solution of class C1,2([0,1) × R) ∩ C0([0,1] × R)
of the following Cauchy problem
?
?t
ψ
S1
S1.
1
2σ2y2∂(2)
v(1,y)
yyv(t,y) + (1 − ry)∂yv(t,y) + ∂tv(t,y)=
=
0,
ψ(y).
on [0,1) × R
Set Zt =
all 0 ≤ t ≤ 1. Then (X0,h) is a self-financing portfolio which replicates the contingent claim
?
Proof. Again for simplicity we will suppose r = 0. We set ξt=Zt
2.10 to the function u(t,z,s) = v(t,z
process v(t,ξt)St,0 ≤ t < 1) as follows:
0Ssds − K, for some K > 0, X0 = v(0,K
??
S0)S0 and ht = v(t,Zt
St) − ∂yv(t,Zt
St)Zt
St, for
1
??1
0Stdt − K
St,0 ≤ t ≤ 1. Applying Proposition
s)s and using the equation fulfilled by v we can expand the
u(t,Zt,St) = v (t,ξt)St= v (0,ξ0)S0+
?t
0
htd−St.
(13)
By arguments which are similar to those used in the proof of Proposition 4.28, it is possible to show
that h is a self-financing portfolio and that (13) implies that u(t,Zt,St) = Xtfor every 0 ≤ t < 1.
Therefore limt→1Xtis finite and equal to ψ (ξ1)S1e−r. This concludes the proof.
4.5. On some sufficient conditions for no-arbitrage
4.5.1. Some illustration on weak geometric Brownian motion
Before we would like to give a first class of non-arbitrage conditions related to the existence of a
A-martingale measure.
For a process X we define the set An
n
?
where 0 = t0< t1< ... < tn= 1 and for every i = 1,...,n
– ui: [0,1] × Ri−→ R of class C1((ti−1,ti) × Ri) ∩ C0([ti−1,ti] × Ri)
– uiand its derivatives have polynomial growth on each interval (ti−1,ti].
Xas the space of all processes h of type:
ht= I[0,t1](t)u1(t,Xt) +
i=2
I(ti−1,ti](t)ui(t,Xt1,...,Xti−1,Xt)
Definition 4.32. A continuous process X, is said weak σ-geometric Brownian motion of
order n if, for every 0 ≤ t0< t1< ... < tn≤ 1
(Xt1,...,Xtn)(P) = law of (Zt1,...,Ztn)
and Z is a weak solution of equation Zt= X0+?·
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0σZ dWt
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28
Remark 4.33. Let n ≥ 4.
1. With the help of Proposition 2.10, examples of such a process can be produced for instance
setting Xt= exp(σBt−σ2t
2. If B is a weak Brownian motion of order n then X is a is a finite quadratic variation process
with [X]t=?t
Similar arguments as in the proof of Proposition 3.17, performed in every subinterval (ti,ti+1],
allow to prove the following.
2) whenever B is a weak Brownian motion of order n.
0σ2X2
tdt.
Proposition 4.34. Suppose that S is a weak σ−geometric Brownian motion of order n with
d[S]t= σ2S2
S-martingale.
tdt. Then S is an An
Definition 4.35. Let Ln
hypotheses of Proposition 4.30 are verified and the process h belongs to An
Corollary 4.36. Suppose that S satisfies the hypotheses of previous proposition. Then the market
is An
Sbe the set of all contingent claims of type ψ(St1,...,Stn) such that the
S.
S-viable and Ln
S-complete.
4.5.2. On some Bender-Sottinen-Valkeila type conditions
The rest of this subsection is inspired by the work of [3] whose results are reformulated below in a
similar but different framework.
For simplicity we will suppose again V ≡ 0 so that the underlying is discounted. We start with
some notations and a definition. Let y0∈ R,t ∈ [0,1]. We denote by Cy0([0,1]) the Banach space
of continuous function η : [0,1] → R such that η(0) = y0. For t ∈ [0,1] we define the shift operator
Θt: C([0,1]) → C([−1,0]) defined by (Θtη)(x) = η(x+t), x ∈ [−1,0]. We remind that continuous
functions defined on some real interval I are naturally prolongated by continuity on the real line.
With a real process S = (St,t ∈ [0,1]) we associate the “window” process St(·) with values in
C([−1,0]), setting St(x) = St+x,x ∈ [−1,0]. S denotes the random element S : Ω −→ C([0,1]),
ω ?→ S(ω).
Definition 4.37. Let Y = (Yt,t ∈ [0,1]) be a process such that Y0= y0for some y0∈ R. Y is said
to fulfill the full support condition if for every η ∈ Cy0([0,1]) one has P{?Y − η?∞≤ ε} > 0.
That notion is present in the classical stochastic analysis literature, see for instance [29]. [23]
introduced a refined version of it which is called the CFS (conditional full support) condition.
Proposition 4.38.
process (σt,t ∈ [0,1]) such that [M]t=?t
condition.
1. Let M be a local martingale such there is a progressively measurable
0σ2
s ∈ [0,1]. We will say in that case that M is a non-degenerate. Then M fulfills the full support
sds,t ∈ [0,1] and a constant c > 0 with σs≥ c,
2. Let G be an independent process from a process M fulfilling the full support condition. Suppose
that G0= 0. Then X = M + G also fulfills the full support condition.
3. Let f : R → R be strictly increasing and continuous. If Y fulfills the full support condition
then f(Y ) also fulfills the support condition.
4. Let Y be a non-degenerate martingale, for instance a Brownian motion, and an independent
process ξ. The process X = eY +ξfulfills the mentioned condition.
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Proof.
1. It is well known that the standard Wiener process fulfills the full support condition.
One possible argument follows directly from a Freidlin-Wentsell type estimate. Given a Brow-
nian motion W, (Wct,t ∈ [0,1]) fulfills the full support condition by a law rescaling argument.
By Dambis, Dubins-Schwarz theorem (see Theorem 1.6 chapter V of [35]), there is a Brown-
ian motion W such that M = W?·
then for any ε > 0,
0σ2
sds. Let η ∈ C0([0,1]); since ?M − η?∞≥ ?Wc·− η?∞,
P{?M − η?∞≤ ε} ≥ P{?Wc·− η?∞≤ ε} > 0
and the result follows.
2. Let g ∈ C0([0,T]) be a realization of G. We set Ψ(g) = P{?M + g − η? ≤ ε}. Clearly
P{?M +G−η? ≤ ε} = E(Ψ(G)). By item 1. Ψ(g) is strictly positive for any g, so the result
follows.
3. Obvious.
4. It follows from previous items.
Assumption 4.39. Let S be a continuous process such that S0= s0and σ : [0,1]×C([−1,0]) −→
R be a continuous functional. Let A be a class of self-financing portfolios h with corresponding
strategies φ =??h0
polynomial growth such that ht= H(t,St(·)) = H(t,ΘtS), t ∈ [0,1[.
We say that A fulfills Assumption 4.39 (with respect to σ) if there is a continuous functional
V = Vφ : C([0,1]) −→ R such that, whenever [S]t =
probability Q, then
?1
In particular the right-hand side forward integral exists with respect to Q.
t,ht
?,0 ≤ t < 1?with associated wealth process Xt(φ) = h0
t+ ht· St. For every
h ∈ A, we suppose the existence of a continuous functional H : [0,1] × C[−1,0] −→ R with
?t
0σ2(s,Ss(·))S2
s,ds with respect to some
Vφ(S) =
0
hsd−Ss
Q a.s.
(14)
We recall that BV ([0,1]) denotes the linear space of bounded variation function f : [0,1] → R
equipped with the topology of weak convergence of the related measures.
Proposition 4.40. Let S be a continuous process such that S0= s0and σ : [0,1]×C([−1,0]) −→ R
be continuous. Let A be constituted by the self-financing portfolios h such there exists a con-
tinuous ϕ : [0,1] × Rn× R −→ R with polynomial growth such that ϕ ∈ C1([0,1[×Rn× R),
(t,v1,...,,vn,x) ?→ ϕ(t,v1,...,,vn,x), and
ht= ϕ?t,V1
H(t,γ) = ϕ?t,V1
where γ ?→ Vi(γ) is continuous from C([−1,0]) to the class of bounded variation functions BV ([0,1]).
Then A fulfills Assumption 4.39 with respect to σ.
Proof. In order to relax the notations we just suppose n = 1. We set ˜ ϕ(t,v,x) =?x
t(St(·)),...,Vn
t(St(·)),St
?= ϕ?t,V1
t(ΘtS),...,Vn
t(ΘtS),St
?, i.e.
t(γ),...,Vn
t(γ),γ(0)?
0ϕ(t,v,y)dy, t ∈
R,v,x ∈ [0,1]. Let Q be a probability under which [S]t=?t
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0σ2(s,Ss(·))ds. By Itô formula Propo-
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30
sition 2.10 applied reversely to ˜ ϕ(t,Yt,St) for Yt:= V1
?1
t(St(·)) from 0 to 1, we get
?1
0
hsd−Ss
=˜ ϕ(1,V1
1(S1(·)),S1) − ˜ ϕ(0,V1
0(S0(·)),S0) −
0
∂s˜ ϕ(s,V1
s(Ss(·)),Ss)ds
(15)
−
1
2
?1
0
∂xϕ(s,V1
s(Ss(·)),Ss)σ2(s,Ss(·))ds −
?1
0
∂v˜ ϕ(s,V1
s(Ss(·)),Ss)dV1
s(Ss(·)).
Setting
V(η)=˜ ϕ(1,V1
1(Θ1η),η(1)) − ˜ ϕ(0,V1
0(Θ0η),η(0)) −
?1
0
∂s˜ ϕ(s,V1
s(Θsη),η(s))ds
(16)
−
1
2
?1
0
∂xϕ(s,V1
s(Θsη),η(s))σ2(s,Ss(·))ds −
?1
0
∂v˜ ϕ(s,Θsη,η(s))dV1
s(Θsη).
The continuity of previous expression is obvious and so Assumption 4.39 is fulfilled.
Examples of classes of strategies which fulfill Assumption 4.39 by Proposition 4.40.
Example 4.41. Let S be a finite quadratic variation such that S0 = s0 for some s0 ∈ R. We
suppose moreover [S]t=?t
1. The class of strategies are determined by ϕ : [0,1] × R −→ R of class C1, (t,x) → ϕ(t,x).
We set H(t,η) = ϕ(t,η(0)). We denote ˜ ϕ(t,x) =
Proposition 2.10 we get
?t
Setting
?t
Assumption 4.39 is verified via Proposition 4.40. The class here defined is a subclass of AS
defined in the introduction.
2. As emphasized in [3], possible choices of Vigiven in Proposition 4.40, are given by V1
minr∈[−t,0]{γ(r)}, V2
minology, we could call those functional Vi, i = 1,2,3, inside factors.
0σ2(s,Ss(·))S2
sds where σ : [0,1] × C([−1,0]) −→ R is continuous with
linear growth.
?x
∂s˜ ϕ(s,Ss)ds −1
0ϕ(t,z)dz. By Itô’s formula given in
0
ϕ(s,Ss)d−Ss= ˜ ϕ(t,St) − ˜ ϕ(0,S0) −
?t
0
2
?t
0
∂xϕ(s,Ss)ds
V(η) = ˜ ϕ(1,η(1)) − ˜ ϕ(0,η(0)) −
0
∂s˜ ϕ(s,Θsη(0))ds −1
2
?t
0
∂xϕ(s,Θsη(0))ds,
t(γ) =
t(γ) = maxr∈[−t,0]{γ(r)}, V3
t(γ) =?0
−tγ(r)dr. According to the [3] ter-
Remark 4.42.
ment of Proposition 4.40 the class of strategies can be enlarged considering similar classes of
strategies on each subinterval ]ti,ti+1].
The class of strategies A constituted by the portfolio strategies h such that for every i there
exists an integer ni≥ 0 such that
htI]ti,ti+1]= ϕi?t,St1,...,Sti,St,V1
for some suitable continuous functions ϕi: [0,T]×Ri×R×Rni−→ R and Vj: C([−1,0]) −→
R, for any 1 ≤ j ≤ ni.
1. Let 0 = t0 < ··· < tn = 1 be a subdivision of [0,1] interval. In the state-
t(St(·)),...,Vni
t (St(·))?
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2. Other classes of strategies fulfilling Assumption 4.39 can be derived through infinite dimen-
sional PDEs, see [15] and [16].
Theorem 4.43. Let s0 > 0, σ : [0,1] × C([−1,0]) → R and two constants c1,c2> 0 such that
c1≤ σ ≤ c2. Suppose the following.
1. The SDE Yt = s0+?t
2. Let S be such that S0= s0and [S]t=?t
3. S fulfills the full support condition with respect to P.
4. Let A be a class of self-financed portfolios h verifying Assumption 4.39.
Then the corresponding market is A-arbitrage free.
Remark 4.44. Of course item 1. may be replaced with the weak existence of the SDE Rt =
logs0+?t
Proof.
Let h ∈ A be a self-financing portfolio and φ = (h0,h) according to Proposition 4.7; let
Xt(φ) be the wealth process such that X0(0) = 0 P-a.s. Without restriction of generality we can
suppose that X1(φ) =?1
suppose X1(φ) ≥ 0 P-a.s. It remains to show that X1(φ) = 0 P-a.s. We denote C := Cs0([0,1]). We
first show that Vφ(η) ≥ 0 for any η ∈ C. For this we suppose ab absurdo that it were not the case.
Then there would exist η0∈ C and ε > 0 such that Vφ(η) < 0 for all η such that ?η − η0?∞≤ ε.
Consequently
0σ(s,Ys(·))YsdWs admits weak strictly positive existence for some
Brownian motion W.
0σ2(s,Ss(·))S2
sds (under the given probability P).
0σ(s,Rs(·))dWsfor some real process R.
0hsd−Ss. In reference to Assumption 4.39, which is verified, we consider
the corresponding continuous functional Vφ: C([0,1]) → R. In particular V(S) = X1(φ) P-a.s. We
P{X1(φ) < 0} = P{V(S) < 0} ≥ P{Vφ(S) < 0;?S − η0?∞≤ ε} > 0.
This contradicts the fact that X1(φ) ≥ 0 P-a.s. It remains to prove that X1(φ) = 0 P-a.s.
By assumption, let¯P a probability under which S is a local martingale with [S]t=?t
By Proposition 4.22 it follows that h cannot be an arbitrage under the probability¯P, if we show
that S is a¯P-A-martingale. This is true whenever
??1
for every hs= H(s,Ss(·)), as in Assumption 4.39. This can be shown using the fact that H has
polynomial growth and σ is bounded. In fact E¯ P
Burkholder-Davis-Gundy inequality and some exponential estimates.
Under¯P we have St= s0eMt+Atwhere Mt=?t
type argument, M +A has the same property. By item 4. of Proposition 4.38 finally also S fulfills
the same condition (under¯P). By a similar reasoning as in the first part of the proof, we obtain
that Vφvanishes identically. Finally X1(φ) = Vφ(S) = 0 P-a.s. and this concludes the proof.
0σ2(s,Ss(·))S2
sds.
By assumption 4.39, Vφ(S) =?1
0hsdSs¯P-a.s. by Proposition 2.7. Consequently X1(φ) ≥ 0¯P-a.s.
E¯ P
0
H2(s,Ss(·))σ2(s,Ss(·))S2
sds
?
< ∞
?supt≤1|St|q?< ∞ for every q > 1 again using
0σ(s,Ss(·))dWsand At= −1
2
?t
0σ2(s,Ss(·))ds. By
item 1. of Proposition 4.38 M fulfills the full support condition with respect to¯P. By a Girsanov
Remark 4.45.
of non-arbitrage, see [13] Theorem 14.1.1. Their notion of non-arbitrage is however a bit
1. Instead of applying Proposition 4.22 we could have used the classical theory
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32
different from ours. In that case one should restrict the class A requiring that the wealth
process associated with h is lower bounded by a (presumably negative) constant in order to
avoid doubling strategies.
2. An interesting question which is beyond the scope of our paper is the following. Suppose that
the underlying S fulfills the full support condition and that Assumption 4.39 is in force. Is
there any A-martingale measure?
5. Utility maximization
5.1. An example of A-martingale and a related optimization problem
We illustrate a setting where Proposition 3.35 applies and it provides a very similar results to
theorem 3.2 of [28]. There, the authors study a particular case of the optimization problem con-
sidered in Proposition 3.35. As process ξ they take a Brownian motion W, and they find sufficient
conditions in order to have existence of a process γ such that W −?·
anticipating setting and combine Malliavin calculus with substitution formulae, the anticipation
being generated by a random variable possibly depending on the whole trajectory of W.
0γtdt is (in our terminology) an
A-martingale, being A some specific set we shall clarify later. To get their goal, they consider an
We work into the specific framework of subsection 4.2.2 .
Assumption 5.1. We suppose the existence of a random variable G in D1,2, satisfying the following
assumption:
?
2. for a.a. t in [0,1] the process
1.
RE
?
|G|2I{0≤x≤G}∪{0≥x≥G}
?
dx < +∞;
I(·,t,G) := I[t,1](·)I{
?1
t(DsG)2ds>0}
??1
t
(DsG)2ds
?−1
(DtG)(D·G)
belongs to Domδ and there exists a P(F)×B(R)-measurable random field (h(t,x),0 ≤ t ≤ 1,x ∈ R)
such that h(·,G) belongs to L2(Ω × [0,1]) and
??1
Let Θ(G) be the set of processes (θt,0 ≤ t < 1) such that there exists a random field (u(t,x),0 ≤
t ≤ 1,x ∈ R) with θt= u(t,G), 0 ≤ t < 1 and
Suppose that A equals Θ(G). With the specifications above we have the following.
E
0
I(u,t,G)dWu| Ft∨ σ(G)
?
= h(t,G),
0 ≤ t ≤ 1.
u(t,·) ∈ C1(R) ∀ 0 ≤ t ≤ 1.
?n
E
R
−n
??
E
?1
0(∂xu(t,x))2dtdx < +∞,∀n ∈ N a.s..
??1
??1
0(∂xu(t,x))2dt
?2
dx +?1
??1
0(u(t,0))2dt
?
< +∞.
???1
0(∂xu(t,G))2(DtG)2dt +
0(∂xu(t,G))2dt
0(DtG)2dt
??
< +∞.
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Corollary 5.2. Let b be a process in L2(Ω × [0,1]), such that h(·,G) + b belongs to the closure of
A in L2(Ω × [0,1]). There exists an optimal process π in A for the function
??1
if and only if h(·,G) + b belongs to A and h(·,G) + b = π.
Proof. It is clear that A is a real linear space of measurable and with bounded paths processes
verifying condition 1. of Assumption 3.29. Proposition 2.8 of [28] shows that every θ in A is in
L2(Ω×[0,1]), that θ is W-improperly forward integrable and that the improper integral belongs to
L2(Ω). In particular, condition 2. of Assumption 3.29 is verified. Furthermore, the proof of theorem
3.2 of [28] implicitly shows that the process W −?·
by Proposition 3.35 setting ξ = W +?·
5.2. Formulation of the problem
θ ?→ E
0
θtd−
?
Wt+
?t
0
bsds
?
−1
2
?1
0
θ2
tdt
?
0h(t,L)dt, is a A-martingale. This implies that
W +?·
0btdt −?·
0γtdt, with γ = h(·,G) + b, is an A-martingale. The end of the proof follows then
0btdt.
We consider the problem of maximization of expected utility from terminal wealth starting from
initial capital X0> 0, being X0a G0-measurable random variable. We define the function U(x)
modeling the utility of an agent with wealth x at the end of the trading period. The function U is
supposed to be of class C2((0,+∞)), strictly increasing, with U′(x)x bounded.
We will need the following assumption.
Assumption 5.3. The utility function U verifies
U
′′(x)x
U′(x)≤ −1,
∀x > 0.
A typical example of function U verifying Assumption 5.3 is U(x) = log(x).
We will focus on portfolios with strictly positive value. As a consequence of this, before starting
analyzing the problem of maximization, we show how it is possible to construct portfolio strategies
when only positive wealth is allowed.
Definition 5.4. For simplicity of calculation we introduce the process
A = log(S) − log(S0) +1
2
?·
0
1
S2
t
d[S]t.
Lemma 5.5. Let θ = (θt,0 ≤ t < 1) be a G-adapted process in C−
1. θ is A-improperly forward integrable.
2. The process Aθ=?·
3. If Xθis the process defined by
b([0,1)) such that
0θsd−Ashas finite quadratic variation.
Xθ= X0exp
??·
0θsd−Asimproperly exist and
?t
0
θtd−At+
?·
0
(1 − θt)dVt−1
2
?Aθ??
,
then?·
0Xθ
tθtd−Atand?·
0Xθ
?·
td−?t
Xθ
0
td−
0
θsd−As=
?·
0
Xθ
tθtd−At
(17)
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Then the couple (X0,h), with ht=
wealth Xθ. In particular, limt→1Xθ
θtXθ
St, 0 ≤ t < 1, is a self-financing portfolio with strictly positive
t= Xθ
t
1exists and it is strictly positive.
Proof. Again, for simplicity we suppose˜S = S therefore V = 0. Thanks to Proposition 2.11 h is
locally S-forward integrable and
2.10, and using hypothesis 3., Xθ=?
Xθ
t= X0+
0
?·
0htd−St =
Xθcan be rewritten in the following way:
?t
?·
0θtXθ
td−At. Applying Corollary 2.9, Proposition
θsd−As= X0+
?t
0
hsd−Ss.
(18)
Proposition 4.7 tells us that Xθis the wealth of the self-financing portfolio (X0,h).
Remark 5.6. The process θ in previous lemma represents the proportion of wealth invested in
S.
Remark 5.7. Let θ be as in Lemma 5.5. Then, for every 0 ≤ t < 1, X is, indeed, the unique
solution, on [0,t], of equation
?·
In fact, uniqueness is insured by Corollary 5.5 of [41]. It is important to highlight that, without
the assumption on θ regarding the chain rule in equality (17), we cannot conclude that Xθsolves
equation (18). However we need to require that Xθsolves the latter equation to interpret it as
the value of a portfolio whose proportion invested in S is constituted by θ. In the sequel we will
construct, in some specific settings, classes of processes defining proportions of wealth as in Lemma
5.5. We will consider, in particular, two cases already contemplated in [5] and [28]. Our definitions
of those sets will result more complicated than the ones defined in the above cited papers. This
happens because, in those works, the chain rule problem arising when the forward integral replaces
the classical Itô integral is not clarified.
Xθ= X0+
0
Xθ
td−
??t
0
θsd−As+
?t
0
(1 − θs)dVs−1
2
?Aθ?
t
?
.
Assumption 5.8. We assume the existence of a real linear space A+of G-adapted processes
(θt,0 ≤ t < 1) in C−
1. θ verifies condition 1., 2. and 3. of Lemma 5.5, and?Aθ?=?·
b([0,1)), such that
0θ2
td[A]t.
2. θI[0,t]belongs to A+for every 0 ≤ t < 1.
For every θ in A+we denote with Qθthe probability measure defined by:
dQθ
dP
=
U′(Xθ
E?U′(Xθ
1)Xθ
1)Xθ
1
1
?.
The utility maximization problem consists in finding a process π in A+maximizing the expected
utility from terminal wealth, i.e.:
θ∈A+E?U(Xθ
Problem (19) is not trivial because of the uncertain nature of the processes A and V and the non
zero quadratic variation of A. Indeed, let us suppose that [A] = 0 and that both A and V are
deterministic. Then, it is sufficient to consider
λ∈RE?U(Xλ
imsart-generic ver. 2007/09/18 file: NSModels9Fev2011Sent.tex date: February 11, 2011
π = arg max
1)?.
(19)
sup
1)?= lim
x→+∞U(x),
Page 35
35
and remind that U is strictly increasing, to see that a maximum can not be realized. The problem
is less clear when the term −1
In the sequel, we will always assume the following.
2
?·
0θ2
td[A]tand a source of randomness are added.
Assumption 5.9. For every θ in A+,
E
?????
?1
0
θtd−(At− Vt)
????+1
2
?1
0
θ2
t[A]t
?
< +∞.
Definition 5.10. A process π is said optimal portfolio in A+, if it is optimal for the function
θ ?→ E?U(Xθ
Remark 5.11. Set ξ = A − V, A = A+, and
?
According to definitions of section 3.3.2, A satisfies Assumption 3.29, the function F is measur-
able, almost surely in C1(R), strictly increasing and with bounded first derivative. If U satisfies
Assumption 5.3 then F is also concave. Moreover F(Lθ) = U(Xθ
1)?in A+, according to definition 3.26.
F(ω,x) = UX0(ω)ex+V1(ω)?
,
(ω,x) ∈ Ω × R.
1) for every θ in A+.
5.3. About some admissible strategies
Before stating some results about the existence of an optimal portfolio, we provide examples of
sets of admissible strategies with positive wealth.
Similarly to section 4.2, it is possible to exhibit classes of admissible strategies fulfilling the corre-
sponding technical assumption. In the context of utility maximization that assumption is Assump-
tion 5.8.
We omit technical details since similar calculations were performed in previous sections. We only
supply precise statements.
1. Admissible strategies via Itô fields. For this example the reader should keep in mind
subsection 4.2.1.
Proposition 5.12. Let A+be the set of all processes (θt,0 ≤ t < 1) such that θ is the
restriction to [0,1) of a process h belonging to C1
Assumption 5.8.
2. Admissible strategies via Malliavin calculus. We restrict ourselves to the setting of
section 4.2.2. We recall that in that case A =
additional assumption:
A(G). Then A+satisfies the hypotheses of
?·
0σtdWt+?·
0µtdt. We make the following
S0= e
?·
0rtdt,
with r in L1,zfor some z > 4 and F-adapted.
Proposition 5.13. Let A+be the set of all G-adapted processes in C−
restriction on [0,1) of processes h in L1,2
variable
?
s∈[0,1]
b([0,1)) being the
−, and the random
−∩ L2,2, such that D−h is in L1,2
sup
t∈[0,1]
|ht| + sup|Dsht| + sup
s,u∈[0,1]
|DsDuht|
?
imsart-generic ver. 2007/09/18 file: NSModels9Fev2011Sent.tex date: February 11, 2011
Page 36
36
is bounded.
Then A+satisfies the hypotheses of Assumption 5.8.
3. Admissible strategies via substitution.
We return here to the framework of subsection 4.2.3.
Proposition 5.14. Let A+be the set of all processes which are the restriction to [0,1)
of processes in Ap,γ(G) for some p > 3 and γ > 3d. Then A+satisfies the hypotheses of
Assumption 5.8.
5.4. Optimal portfolios and A+-martingale property
Adapting results contained in section 3.3.2 to the utility maximization problem, we can formulate
the following propositions. We omit their proofs, being particular cases of the ones contained in
that section.
Proposition 5.15. If a process π in A+is an optimal portfolio, then the process A−V −?·
Proposition 5.16. Suppose that A+satisfies Assumption D (see Definition 3.6) with respect to G.
If a process π in A+is an optimal portfolio, then the process A−V −?·
β = π +
pπ
d[A]
0πtd[A]t
is an A+-martingale under Qπ. If U fulfills Assumption 5.3, then the converse holds.
0βtd[A]tis a G-martingale
?
under P, with
1
d[pπ,A]
,
and
pπ= EQπ?dP
dQπ| G·
.
If U fulfills Assumption 5.3, then the converse holds.
Remark 5.17.
pearing in Propositions 5.15 and 5.16 is equal to P.
1. We emphasize that if U(x) = log(x), then the probability measure Qπap-
2. In [2] it is proved that if the maximum of expected logarithmic utility over all simple admissible
strategies is finite, then S is a semimartingale with respect G. This result does not imply
Proposition 5.16. Indeed, we do not need to assume that our set of portfolio strategies contains
the set of simple predictable admissible ones. On the contrary, we want to point out that, as
soon as the class of admissible strategies is not large enough, the semimartingale property of
price processes could fail, even under finite expected utility.
Proposition 5.18. Suppose that U(x) = log(x), x in (0,+∞). Assume that there exists a mea-
surable process γ such that A − V −?·
2. Suppose moreover that there exists a sequence (θn)n∈N⊂ A+such that
??1
If an optimal portfolio π exists, then d[A]{t ∈ [0,1),πt?= γt} = 0 almost surely.
0γtd[A]tis an A+-martingale.
1. If γ belongs to A+then it is an optimal portfolio.
lim
n→+∞E
0
|θn
t− γt|2d[A]t
?
= 0.
5.5. Example
We adopt the setting of section 2 and we further assume that σ is a strictly positive real.
imsart-generic ver. 2007/09/18 file: NSModels9Fev2011Sent.tex date: February 11, 2011
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37
Proposition 5.19. If a process π is an optimal portfolio in A+, then the process W−?·
0
?rt−µt
σ
+ πtσ?dt
is an A+-martingale under Qπ. If U fulfills Assumption 5.3, then the converse holds.
Proof. First of all we observe that it is not difficult to prove that A+satisfies Assumption 5.9. If
a process π is an optimal portfolio in A+then Proposition 5.15 implies that the process Mπ, with
Mπ= σ?W −?·
0
σ
Similarly, if U satisfies Assumption 5.3, the converse follows by Proposition 5.15.
0
?rt−µt
σ
− πtσ?dt?, is an A+-martingale under Qπ. We observe that σ−1A+= A+.
Therefore, σ−1Mπ= W −?·
?rt−µt
+ πtσ?dt is an A+-martingale.
Corollary 5.20. Let A+satisfy Assumption D with respect to G. If a process π in A+is an
optimal portfolio then the process B = W −?·
α = πσ +r − µ
σpπ
d[W]
0αtdt with
+
1
d[pπ,W]
,
and
pπ= EQπ?dP
dQπ| G·
?
,
is a G-Brownian motion under P. If U satisfies Assumption 5.3, then the converse holds.
Proof. Let π be an optimal portfolio. By Proposition 3.34, the process B is a G-martingale and so
a G-Brownian motion under P.
The results concerning the example above were proved in [5]. We generalize them in two directions:
we replace the geometric Brownian motion A by a finite quadratic variation process and we let the
set of possible strategies vary in sets which can, a priori, exclude some simple predictable processes.
5.6. Example
We consider the example treated in section 5.1. We suppose, for simplicity, that
St= S0eσWt+
?
µ−σ2
2
?
t,S0
t= ert
0 ≤ t ≤ 1,
being σ, µ and r positive constants. This implies At= σWt+ µt, and Vt= rt for 0 ≤ t ≤ 1. We
set A+= Θ(L).
Proposition 5.21. Suppose that U(x) = log(x), x in (0,+∞). Suppose that h(·,L) belongs to
the closure of Θ(L) in L2(Ω × [0,1]). Then an optimal portfolio π exists if and only if the process
h(·,L) +?·
Proof. The result follows from Corollary 5.2.
0
µ−r
σdt belongs to Θ(L) and π = h(·,L) +µ−r
σ.
Sufficiency for the Proposition above was shown, with more general σ, r and µ in theorem 3.2
of [28]. Nevertheless, in this paper we go further in the analysis of utility maximization problem.
Indeed, besides observing that the converse of that theorem holds true, we find that the existence
of an optimal strategy is strictly connected, even for different choices of the utility function, to the
A+-semimartingale property of W. To be more precise, in that paper the authors show that an
optimal process exists, under the given hypotheses, handling directly the expression of the expected
utility, which has, in the logarithmic case, a nice expression. Here we reinterpret their techniques
at a higher level which permits us to partially generalize those results.
imsart-generic ver. 2007/09/18 file: NSModels9Fev2011Sent.tex date: February 11, 2011
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