# On stochastic calculus related to financial assets without semimartingales

**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

$\mathcal{A}$ of admissible strategies is restricted. The classical notion of

martingale is replaced with the notion of $\mathcal{A}$-martingale. A calculus

related to $\mathcal{A}$-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.

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**ABSTRACT:**This article focuses on a new concept of quadratic variation for processes taking values in a Banach space $B$ and a corresponding covariation. This is more general than the classical one of M\'etivier and Pellaumail. Those notions are associated with some subspace $\chi$ of the dual of the projective tensor product of $B$ with itself. We also introduce the notion of a convolution type process, which is a natural generalization of the It\^o process and the concept of $\bar \nu_0$-semimartingale, which is a natural extension of the classical notion of semimartingale. The framework is the stochastic calculus via regularization in Banach spaces. Two main applications are mentioned: one related to Clark-Ocone formula for finite quadratic variation processes; the second one concerns the probabilistic representation of a Hilbert valued partial differential equation of Kolmogorov type.Metrika 01/2013; · 0.72 Impact Factor - SourceAvailable from: export.arxiv.org[Show abstract] [Hide abstract]

**ABSTRACT:**In this paper we study the forward integral of operator-valued processes with respect to a cylindrical Brownian motion. In particular, we provide conditions under which the approximating sequence of processes of the forward integral, converges to the stochastic integral process with respect to Sobolev norms of smoothness alpha < 1/2. This result will be used to derive a new integration by parts formula for the forward integral.10/2013;

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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

1

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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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9

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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10

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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11

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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15

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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