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Functional Ito calculus and stochastic integral representation

of martingales

Rama Cont David-Antoine Fournie

First draft: June 2009. This version: June 2010.∗

Abstract

We develop a non-anticipative calculus for functionals of a continuous semimartingale, using

a notion of pathwise functional derivative. A functional extension of the Ito formula is derived

and used to obtain a constructive martingale representation theorem for a class of continuous

martingales verifying a regularity property. By contrast with the Clark-Haussmann-Ocone for-

mula, this representation involves non-anticipative quantities which can be computed pathwise.

These results are used to construct a weak derivative acting on square-integrable martingales,

which is shown to be the inverse of the Ito integral, and derive an integration by parts formula for

Ito stochastic integrals. We show that this weak derivative may be viewed as a non-anticipative

“lifting” of the Malliavin derivative.

Regular functionals of an Ito martingale which have the local martingale property are char-

acterized as solutions of a functional differential equation, for which a uniqueness result is given.

Keywords: stochastic calculus, functional calculus, Ito formula, integration by parts, Malliavinderivative, martingale representation, semimartingale, Wiener functionals, functional Feynman-Kacformula, Kolmogorov equation, Clark-Ocone formula.

∗We thank Bruno Dupire for sharing his original ideas with us. R. Cont thanks Hans Follmer, Jean Jacod and the

late Paul Malliavin for helpful comments and discussions.

1

Contents

1 Introduction 3

2 Functionals representation of non-anticipative processes 4

2.1 Horizontal and vertical perturbation of a path . . . . . . . . . . . . . . . . . . . . . . 62.2 Regularity for non-anticipative functionals . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Measurability properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 Pathwise derivatives of non-anticipative functionals 8

3.1 Horizontal and vertical derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.2 Obstructions to regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.3 Uniqueness results for vertical derivatives . . . . . . . . . . . . . . . . . . . . . . . . 11

4 Functional Ito calculus 16

4.1 Functional Ito formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.2 Intrinsic nature of the vertical derivative . . . . . . . . . . . . . . . . . . . . . . . . . 19

5 Stochastic integral representation of martingales 20

5.1 A martingale representation formula . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.2 Relation with the Malliavin derivative . . . . . . . . . . . . . . . . . . . . . . . . . . 21

6 Weak derivatives and integration by parts for stochastic integrals 23

7 Functional equations for martingales 26

A Proof of Theorem 2.6 31

2

1 Introduction

Ito’s stochastic calculus [15, 16, 8, 24, 20, 28] has proven to be a powerful and useful tool in analyzingphenomena involving random, irregular evolution in time.

Two characteristics distinguish the Ito calculus from other approaches to integration, which mayalso apply to stochastic processes. First is the possibility of dealing with processes, such as Brownianmotion, which have non-smooth trajectories with infinite variation. Second is the non-anticipativenature of the quantities involved: viewed as a functional on the space of paths indexed by time, anon-anticipative quantity may only depend on the underlying path up to the current time. Thisnotion, first formalized by Doob [9] in the 1950s via the concept of a filtered probability space, isthe mathematical counterpart to the idea of causality.

Two pillars of stochastic calculus are the theory of stochastic integration, which allows to

define integrals∫ T

0Y dX for of a large class of non-anticipative integrands Y with respect to a

semimartingale X = (X(t), t ∈ [0, T ]), and the Ito formula [15, 16, 24] which allows to representsmooth functions Y (t) = f(t,X(t)) of a semimartingale in terms of such stochastic integrals. Acentral concept in both cases is the notion of quadratic variation [X ] of a semimartingale, whichdifferentiates Ito calculus from the calculus of smooth functions. Whereas the class of integrands Ycovers a wide range of non-anticipative path-dependent functionals of X , the Ito formula is limitedto functions of the current value of X .

Given that in many applications such as statistics of processes, physics or mathematical finance,one is led to consider functionals of a semimartingale X and its quadratic variation process [X ] suchas:

∫ t

0

g(t,Xt)d[X ](t), G(t,Xt, [X ]t), or E[G(T,X(T ), [X ](T ))|Ft] (1)

(where X(t) denotes the value at time t and Xt = (X(u), u ∈ [0, t]) the path up to time t) there hasbeen a sustained interest in extending the framework of stochastic calculus to such path-dependentfunctionals.

In this context, the Malliavin calculus [3, 4, 25, 23, 26, 29, 30] has proven to be a powerful toolfor investigating various properties of Brownian functionals, in particular the smoothness of theirdensities. Since the construction of Malliavin derivative, which is a weak derivative for functionalson Wiener space, does not refer to an underlying filtration Ft, it naturally leads to representationsof functionals in terms of anticipative processes [4, 14, 26]. However, in many applications it is morenatural to consider non-anticipative versions of such representations.

In a recent insightful work, B. Dupire [10] has proposed a method to extend the Ito formula toa functional setting in a non-anticipative manner. Building on this insight, we develop hereafter anon-anticipative calculus [6] for a class of functionals -including the above examples- which may berepresented as

Y (t) = Ft(X(u), 0 ≤ u ≤ t, A(u), 0 ≤ u ≤ t) = Ft(Xt, At) (2)

where A is the local quadratic variation defined by [X ](t) =∫ t

0A(u)du and the functional

Ft : D([0, t],Rd)×D([0, t], S+d ) → R

represents the dependence of Y on the path of X and its quadratic variation. For such functionals,we define an appropriate notion of regularity (Section 2.2) and a non-anticipative notion of pathwise

3

derivative (Section 3). Introducing At as additional variable allows us to control the dependenceof Y with respect to the ”quadratic variation” [X ] by requiring smoothness properties of Ft withrespect to the variable At in the supremum norm, without resorting to p-variation norms as in roughpath theory [21]. This allows to consider a wide range of functionals, including the examples (1).

Using these pathwise derivatives, we derive a functional Ito formula (Section 4), which extendsthe usual Ito formula in two ways: it allows for path-dependence and for dependence with respect toquadratic variation process (Theorem 4.1). This result gives a rigorous mathematical framework fordeveloping and extending the ideas proposed by B. Dupire [10] to a large class of functionals whichnotably includes stochastic integrals and allows for dependence on the quadratic variation along apath.

We use the functional Ito formula to derive a constructive version of the martingale representationtheorem (Section 5), which can be seen as a non-anticipative counterpart of the Clark-Haussmann-Ocone formula [4, 13, 14, 26].

The martingale representation formula allows to obtain an integration by parts formula for Itostochastic integrals (Theorem 6.4), which enables in turn to define a weak functional derivative fora class of square-integrable martingales (Section 6). We argue that this weak derivative may beviewed as a non-anticipative “lifting” of the Malliavin derivative (Theorem 6.9).

Finally, we show that regular functionals of an Ito martingale which have the local martingaleproperty are characterized as solutions of a functional analogue of Kolmogorov’s backward equation(Section 7), for which a uniqueness result is given (Theorem 7.2).

Sections 2, 3 and 4 are essentially “pathwise” results which may be restated in purely analyticalterms [5], following H. Follmer’s [12] pathwise approach to Ito calculus. Sections 5, 6 and 7 useIto’s stochastic integral in a crucial way to define a weak functional calculus for square-integrablemartingales.

2 Functionals representation of non-anticipative processes

Let X : [0, T ]×Ω 7→ Rd be a continuous, Rd−valued semimartingale defined on a filtered probabilityspace (Ω,B,Bt,P). The paths of X then lie in C0([0, T ],R

d), which we will view as a subspace ofD([0, t],Rd) the space of cadlag functions with values in Rd. For a path x ∈ D([0, T ],Rd), denoteby x(t) the value of x at t and by xt = (x(u), 0 ≤ u ≤ t) the restriction of x to [0, t]. Thus xt ∈D([0, t],Rd). For a processX we shall similarly denoteX(t) its value at t andXt = (X(u), 0 ≤ u ≤ t)its path on [0, t].

Denote by Ft = FXt+ the right-continuous augmentation of the natural filtration of X and by P

(resp. O) the predictable (resp. optional) sigma-algebra on [0, T ]. Denote by [X ] = ([X i, Xj], i, j =1..d) the quadratic (co-)variation process associated to X , taking values in the set S+

d of positived× d matrices. We assume that

[X ](t) =

∫ t

0

A(s)ds (3)

for some cadlag process A with values in S+d . The paths of A lie in St = D([0, t], S+

d ), the space ofcadlag functions with values S+

d .A process Y : [0, T ] × Ω 7→ Rd which is progressively measurable with respect to Ft may be

represented as

Y (t) = Ft(X(u), 0 ≤ u ≤ t, A(u), 0 ≤ u ≤ t) = Ft(Xt, At) (4)

4

where F = (Ft)t∈[0,T ] is a family of functionals

Ft : D([0, t],Rd)× St → R

representing the dependence of Y (t) on the underlying path of X and its quadratic variation.Introducing the process A as additional variable may seem redundant at this stage: indeed

A(t) is itself Ft− measurable i.e. a functional of Xt. However, it is not a continuous functionalwith respect to the supremum norm or other usual topologies on D([0, t],Rd). Introducing At asa second argument in the functional will allow us to control the regularity of Y with respect to[X ]t =

∫ t

0 A(u)du simply by requiring continuity of Ft in supremum or Lp norms with respect tothe “lifted process” (X,A) (see Section 2.2). This idea is analogous in some ways to the approachof rough path theory [21], although here we do not resort to p-variation norms.

Since Y is non-anticipative, Y (t, .) only depends on the path up to t. Thus, we can also viewF = (Ft)t∈[0,T ] as a map defined on the vector bundle:

Υ =⋃

t∈[0,T ]

D([0, t],Rd)×D([0, t], S+d ) (5)

This motivates the following definition:

Definition 2.1 (Non-anticipative functional on path space). A non-anticipative functional on Υ isa family F = (Ft)t∈[0,T ] where

Ft : D([0, t],Rd)×D([0, t], S+d ) 7→ R

(x, v) → Ft(x, v)

is measurable with respect to Bt, the filtration generated by the canonical process on D([0, t],Rd)×D([0, t], S+

d ).

We further denote

Υc =⋃

t∈[0,T ]

C([0, t],Rd)×D([0, t], S+d ) (6)

If Y is a predictable (P−measurable) process, then [8, Vol. I,Par. 97]

∀t ∈ [0, T ], Y (t, ω) = Y (t, ωt−)

where ωt− denotes the path defined on [0, t] by

ωt−(u) = ω(u) u ∈ [0, t[ ωt−(t) = ω(t−)

Note that ωt− is cadlag and should not be confused with the caglad path u 7→ ω(u−).The functionals discussed in the introduction depend on the process A via [X ] =

∫ .

0 A(t)dt. Inparticular, they satisfy the condition Ft(Xt, At) = Ft(Xt, At−). Accordingly, we consider throughoutthe paper non-anticipative functionals

F = (Ft)t∈[0,T ] Ft : D([0, t],Rd)× St → R

where F has a “predictable” dependence with respect to the second argument:

∀t ∈ [0, T ], ∀(x, v) ∈ D([0, t],Rd)× St, Ft(xt, vt) = Ft(xt, vt−) (7)

5

0 0.2 0.4 0.6 0.8 1 1.2−0.5

0

0.5

1

1.5

2

2.5

3

t0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−0.5

0

0.5

1

1.5

2

2.5

3

Figure 1: Left: horizontal extension xt,h of a path x ∈ C0([0, t],R). Right: vertical extension xht .

2.1 Horizontal and vertical perturbation of a path

Consider a path x ∈ D([0, T ]),Rd) and denote by xt ∈ D([0, t],Rd) its restriction to [0, t] for t < T .For h ≥ 0, the horizontal extension xt,h ∈ D([0, t+ h],Rd) of xt to [0, t+ h] is defined as

xt,h(u) = x(u) u ∈ [0, t] ; xt,h(u) = x(t) u ∈]t, t+ h] (8)

For h ∈ Rd, we define the vertical perturbation xht of xt as the cadlag path obtained by shifting theendpoint by h:

xht (u) = xt(u) u ∈ [0, t[ xht (t) = x(t) + h (9)

or in other words xht (u) = xt(u) + h1t=u.We now define a distance between two paths, not necessarily defined on the same time interval.

For T ≥ t′ = t+ h ≥ t ≥ 0, (x, v) ∈ D([0, t],Rd)× S+t and (x′, v′) ∈ D([0, t+ h],Rd)× St+h define

d∞( (x, v), (x′, v′) ) = supu∈[0,t+h]

|xt,h(u)− x′(u)|+ supu∈[0,t+h]

|vt,h(u)− v′(u)|+ h (10)

If the paths (x, v), (x′, v′) are defined on the same time interval, then d∞((x, v), (x′, v′)) is simplythe distance in supremum norm.

2.2 Regularity for non-anticipative functionals

Using the distances defined above, we now introduce classes of (left) continuous functionals on Υ.

Definition 2.2 (Continuity at fixed times). A functional F defined on Υ is said to be continuousat fixed times for the d∞ metric if and only if:

∀t ∈ [0, T [, ∀ǫ > 0, ∀(x, v) ∈ D([0, t],Rd)× St, ∃η > 0, (x′, v′) ∈ D([0, t],Rd)× St,

d∞((x, v), (x′, v′)) < η ⇒ |Ft(x, v)− Ft(x′, v′)| < ǫ (11)

6

Definition 2.3 (Left-continuous functionals). Define C0,0l ([0, T [) as the set of functionals F =

(Ft, t ∈ [0, T [) on Υ which are continuous at fixed times in the sense of Definition 2.2 and ”left-continuous” in t the following sense:

∀t ∈ [0, T [, ∀ǫ > 0,

∀(x, v) ∈ D([0, t],Rd)× St, ∃η > 0, ∀h ∈ [0, t], ∀(x′, v′) ∈ Ut−h × St−h,

d∞((x, v), (x′, v′)) < η ⇒ |Ft(x, v) − Ft−h(x′, v′)| < ǫ (12)

We call a functional “boundedness preserving” if it is bounded on each bounded set of paths:

Definition 2.4 ( Boundedness-preserving functionals). Define B([0, T )) as the set of functionals Fon Υ([0, T ]) such that for every compact subset K of Rd, every R > 0 and t0 < T there exists aconstant CK,R,t0 such that:

∀t ≤ t0, ∀(x, v) ∈ D([0, t],K)× St, sups∈[0,t]

|v(s)| < R⇒ |Ft(x, v)| < CK,R,t0 (13)

2.3 Measurability properties

Composing a non-anticipative functional F with the process (X,A) yields an Ft−adapted processY (t) = Ft(Xt, At). The results below link the measurability and pathwise regularity of Y to theregularity of the functional F .

Lemma 2.5 (Pathwise regularity). If F ∈ C0,0l then for any (x, v) ∈ D([0, T ],Rd) × ST , the path

t 7→ Ft(xt−, vt−) is left-continuous.

Proof. Let F ∈ C0,0l and t ∈ [0, T ). For h > 0 sufficiently small,

d∞((xt−h, vt−h), (xt−, vt−)) = supu∈(t−h,t)

|x(u)− x(t−)|+ supu∈(t−h,t)

|v(u)− v(t−)|+ h (14)

Since x and v are cadlag, this quantity converges to 0 as h→ 0+, so

Ft−h(xt−h, vt−h)− Ft(xt−, vt−)h→0+→ 0

so t 7→ Ft(xt−, vt−) is left-continuous.

Theorem 2.6. • If F is continuous at fixed times, then the process Y defined by Y ((x, v), t) =Ft(xt, vt) is adapted.

• If F ∈ C0,0l ([0, T [), then the process Z(t) = Ft(Xt, At) is optional.

• If F ∈ C0,0l ([0, T [), and if either A is continuous or F has predictable dependence in the second

variable, then the process Z(t) = Ft(Xt, At) is predictable.

7

In particular, any F ∈ C0,0l is a non-anticipative functional in the sense of Definition 2.1.

We propose an easy-to-read proof of points 1. and 3. in the case where A is continuous. The(more technical) proof for the cadlag case is given in the Appendix A.

Continuous case. Assume that F is continuous at fixed times and that the paths of (X,A) are almost-surely continuous. Let’s prove that the paths of Y are adapted: Xt is Ft-measurable. Introduce thesubdivision tin = iT

2n , i = 0..2n of [0, T ], as well as the following piecewise-constant approximationsof X and A:

Xn(t) =

2n∑

k=0

X(tnk )1[tnk ,tnk+1)(t) +XT 1T(t)

An(t) =

2n∑

k=0

A(tnk )1[tnk ,tnk+1)(t) +XT 1T(t) (15)

The random variable Y n(t) = Ft(Xnt , A

nt ) is a continuous function of the random variables

X(tnk), A(tnk ), t

nk ≤ t hence is Ft-measurable. The representation above shows in fact that Y n(t) is

Ft-measurable. Xnt and Ant converge respectively to Xt and At almost-surely so Y n(t)→n→∞ Y (t)

a.s., hence Y (t) is Ft-measurable.Point 1. implies point 3. since the path of Z are left-continuous by Lemma 2.5.

3 Pathwise derivatives of non-anticipative functionals

3.1 Horizontal and vertical derivatives

We now define pathwise derivatives for a functional F = (Ft)t∈[0,T [ ∈ C0,0, following Dupire [10].

Definition 3.1 (Horizontal derivative). The horizontal derivative at (x, v) ∈ D([0, t],Rd) × St ofnon-anticipative functional F = (Ft)t∈[0,T [ is defined as

DtF (x, v) = limh→0+

Ft+h(xt,h, vt,h)− Ft(x, v)

h(16)

if the corresponding limit exists. If (16) is defined for all (x, v) ∈ Υ the map

DtF : D([0, t],Rd)× St 7→ Rd

(x, v) → DtF (x, v) (17)

defines a non-anticipative functional DF = (DtF )t∈[0,T ], the horizontal derivative of F .

Note that our definition (16) is different from the one in [10] where the case F (x, v) = G(x) isconsidered.

Dupire [10] also introduced a pathwise spatial derivative for such functionals, which we nowintroduce. Denote (ei, i = 1..d) the canonical basis in Rd.

8

Definition 3.2. A non-anticipative functional F = (Ft)t∈[0,T [ is said to be vertically differentiable

at (x, v) ∈ D([0, t]),Rd)×D([0, t], S+d ) if

Rd 7→ R

e → Ft(xet , vt)

is differentiable at 0. Its gradient at 0

∇xFt (x, v) = (∂iFt(x, v), i = 1..d) where ∂iFt(x, v) = limh→0

Ft(xheit , v)− Ft(x, v)

h(18)

is called the vertical derivative of Ft at (x, v). If (18) is defined for all (x, v) ∈ Υ, the maps

∇xF : D([0, t],Rd)× St 7→ Rd

(x, v) → ∇xFt(x, v) (19)

define a non-anticipative functional ∇xF = (∇xFt)t∈[0,T ], the vertical derivative of F . F is thensaid to be vertically differentiable on Υ.

Remark 3.3. ∂iFt(x, v) is simply the directional derivative of Ft in direction (1tei, 0). Note thatthis involves examining cadlag perturbations of the path x, even if x is continuous.

Remark 3.4. If Ft(x, v) = f(t, x(t)) with f ∈ C1,1([0, T [×Rd) then we retrieve the usual partialderivatives:

DtF (x, v) = ∂tf(t,X(t)) ∇xFt(Xt, At) = ∇xf(t,X(t)).

Remark 3.5. Bismut [3] considered directional derivatives of functionals on D([0, T ],Rd) in thedirection of purely discontinuous (e.g. piecewise constant) functions with finite variation, which issimilar to Def. 3.2. This notion, used in [3] to derive an integration by parts formula for pure-jumpprocesses, seems natural in that context. We will show that the directional derivative (18) alsointervenes naturally when the underlying process X is continuous, which is less obvious.

Note that, unlike the definition of a Frechet derivative in which F is perturbed along all direc-tions in C0([0, T ],R

d) or the case of a Malliavin derivative [22, 23] in which F is perturbed along allCameron-Martin (i.e. absolutely continuous) functions, we only examine local perturbations, so∇xFand DtF seem to contain less information on the behavior of the functional F . Nevertheless we willshow in Section 4 that these derivatives are sufficient to reconstitute the path of Y (t) = Ft(Xt, At):the pieces add up to the whole.

Definition 3.6. Define Cj,k as the set of functionals F ∈ C0,0l which are differentiable j times

horizontally and k time vertically at all (x, v) ∈ D([0, t],Rd) × St, t < T , such that the horizon-tal derivatives DmF,m ≤ j define functionals which are continuous at fixed times and the verticalderivatives ∇n

xF, n ≤ k are elements of C0,0l ([0, T [) .

Define Cj,kb as the set of functionals F ∈ Cj,k([0, T ]) such that the horizontal derivatives up to

order j and vertical derivatives up to order k are in B.

9

Example 1 (Smooth functions). In the case where F reduces to a smooth function of X(t),

Ft(xt, vt) = f(t, x(t)) (20)

where f ∈ Cj,k([0, T ]× Rd), the pathwise derivatives reduces to the usual ones: F ∈ C

j,kb with:

DitF (xt, vt) = ∂itf(t, x(t)) ∇m

x Ft(xt, vt) = ∂mx f(t, x(t)) (21)

In fact to have F ∈ Cj,k we simply require f to be j times right-differentiable in the time variable,with right-derivatives in t which are continuous in the space variable and f,∇f and ∇2f to be jointlyleft-continuous in t and continuous in the space variable.

Example 2 (Integrals with respect to quadratic variation). A process Y (t) =∫ t

0g(X(u))d[X ](u)

where g ∈ C0(Rd) may be represented by the functional

Ft(xt, vt) =

∫ t

0

g(x(u))v(u)du (22)

It is readily observed that F ∈ C1,∞b , with:

DtF (xt, vt) = g(x(t))v(t) ∇jxFt(xt, vt) = 0 (23)

Example 3. The martingale Y (t) = X(t)2 − [X ](t) is represented by the functional

Ft(xt, vt) = x(t)2 −

∫ t

0

v(u)du (24)

Then F ∈ C1,∞b with:

DtF (x, v) = −v(t) ∇xFt(xt, vt) = 2x(t)

∇2xFt(xt, vt) = 2 ∇j

xFt(xt, vt) = 0, j ≥ 3 (25)

Example 4 (Doleans exponential). The exponential martingale Y = exp(X − [X ]/2) may be repre-sented by the functional

Ft(xt, vt) = ex(t)−12

∫t

0v(u)du (26)

Elementary computations show that F ∈ C1,∞b with:

DtF (x, v) = −1

2v(t)Ft(x, v) ∇j

xFt(xt, vt) = Ft(xt, vt) (27)

Note that, although At may be expressed as a functional of Xt, this functional is not continuousand without introducing the second variable v ∈ St, it is not possible to represent Examples 2, 3and 4 as a left-continuous functional of x alone.

10

3.2 Obstructions to regularity

It is instructive to observe what prevents a functional from being regular in the sense of Definition3.6. The examples below illustrate the fundamental obstructions to regularity:

Example 5 (Delayed functionals). Ft(xt, vt) = x(t − ǫ) defines a C0,∞b functional. All vertical

derivatives are 0. However, it fails to be horizontally differentiable.

Example 6 (Jump of x at the current time). Ft(xt, vt) = x(t) − x(t−) defines a functional which isinfinitely differentiable and has regular pathwise derivatives:

DtF (xt, vt) = 0 ∇xFt(xt, vt) = 1 (28)

However, the functional itself fails to be C0,0l .

Example 7 (Jump of x at a fixed time). Ft(xt, vt) = 1t≥t0(x(t0) − x(t0−)) defines a functional in

C0,0l which admits horizontal and vertical derivatives at any order at each point (x, v). However,

∇xFt(xt, vt) = 1t=t0 fails to be left continuous so F is not C0,1 in the sense of Definition 3.2.

Example 8 (Maximum). Ft(xt, vt) = sups≤t x(s) is C0,0l but fails to be vertically differentiable on

the set(xt, vt) ∈ D([0, t],Rd)× St, x(t) = sup

s≤tx(s).

3.3 Uniqueness results for vertical derivatives

Consider now an Ft−adapted process (Y (t))t∈[0,T ] given by a functional representation

Y (t) = Ft(Xt, At) (29)

where F ∈ C0,0l has left-continuous horizontal and vertical derivatives DtF ∈ C

0,0l and ∇xF ∈ C

0,0l .

SinceX has continuous paths, Y only depends on the restriction of F to Υc =⋃

t∈[0,T ] C([0, t],Rd)×

St. Therefore, the representation (29) of Y by F : Υ → R in (29) is not unique, as the followingexample shows.

Example 9 (Non-uniqueness of functional representation). Take d = 1. The quadratic variationprocess [X ] may be represented by the following functionals:

F 0t (xt, vt) =

∫ t

0

v(u)du

F 1t (xt, vt) =

(

lim supn

t2n∑

i=0

|x(i + 1

2n)− x(

i

2n)|2

)

1limn

∑i≤t2n (x( i+1

2n )−x( i2n ))2<∞

F 2t (xt, vt) =

lim supn

t2n∑

i=0

|x(i + 1

2n)− x(

i

2n)|2 −

∑

0≤s<t

|∆x(s)|2

1lim supn

∑t2n

i=0 |x( i+12n )−x( i

2n )|2<∞ 1∑s<t

|∆x(s)|2<∞

where ∆x(t) = x(t)−x(t−) denotes the discontinuity of x at t. If X is a continuous semimartingale,then

F 0t (Xt, At) = F 1

t (Xt, At) = F 2t (Xt, At) = [X ](t)

Yet F 0 ∈ C1,2b but F 1, F 2 are not even continuous at fixed times, so F i /∈ C

0,0l for i = 1, 2.

11

However, the definition of ∇xF (Definition 3.2), which involves evaluating F on cadlag paths,seems to depend on the values taken by F outside Υc.

The following key result shows that, if a functional F belongs to C1,1, then ∇xFt(Xt, At) isuniquely determined by the value taken by F on continuous paths:

Theorem 3.7. If F 1, F 2 ∈ C1,1, such that DF 1,DF 2 ∈ B, coincide on continuous paths:

∀t < T, ∀(x, v) ∈ C0([0, T ],Rd)× ST , F 1

t (xt, vt) = F 2t (xt, vt)

then ∀t < T, ∀(x, v) ∈ C0([0, T ],Rd)× ST ,∇xF

1t (xt, vt) = ∇xF

2t (xt, vt)

Proof. Let F = F 1 − F 2 ∈ C1,1 and (x, v) ∈ C0([0, T ],Rd) × ST . Then Ft(x, v) = 0 for all t ≤ T .

It is then obvious that DtF (x, v) is also 0 on continuous paths because the extension (xt,h) of xtis itself a continuous path. Assume now that there exists some (x, v) ∈ C0([0, T ],R

d) × ST suchthat for some 1 ≤ i ≤ d and t0 ∈ [0, T [, ∂iFt0(xt0 , vt0−) > 0. Let α = 1

2∂iFt0(xt0 , vt0−). By theleft-continuity of ∂iF and DtF at (xt0 , vt0), we may choose ǫ < T − t sufficiently small such that,for any t′ ∈ [0, t0], for any (x′, v′) ∈ D([0, t′],Rd)× St′ ,

d∞((xt0 , vt0), (x′, v′)) < ǫ⇒ ∂iFt′(x

′, v′) > α and |DtF (x′, v′)| < 1 (30)

Chose t < t0 such that d∞((xt, vt), (xt0 , vt0−)) <ǫ2 , define h := t0 − t and define the following

extension of xt to [0, T ]:

z(u) = x(u), u ≤ t

zj(u) = xj(t) + 1i=j(u− t), t ≤ u ≤ T, 1 ≤ j ≤ d (31)

Define the following sequence of piecewise constant approximations of zt+h:

zn(u) = zn = z(u), u ≤ t

znj (u) = xj(t) + 1i=jh

n

n∑

k=1

1 khn

≤u−t, t ≤ u ≤ t+ h, 1 ≤ j ≤ d (32)

Since d∞((z, vt,h), (zn, vt,h)) =

hn → 0,

|Ft+h(z, vt,h)− Ft+h(zn, vt,h)|

n→+∞→ 0

We can now decompose Ft+h(zn, vt,h)− Ft(x, v) as

Ft+h(zn, vt,h)− Ft(x, v) =

n∑

k=1

Ft+ khn(znt+ kh

n

, vt, khn)− Ft+ kh

n(znt+ kh

n−, vt, kh

n)

+n∑

k=1

Ft+ khn(znt+ kh

n−, vt, kh

n)− F

t+ (k−1)hn

(znt+

(k−1)hn

, vt, (k−1)h

n

) (33)

where the first sum corresponds to jumps of zn at times t+ khn and the second sum to its extension

by a constant on [t+ (k−1)hn , t+ kh

n ].

Ft+ khn(znt+ kh

n

, vt, khn)− Ft+ kh

n(znt+ kh

n−, vt, kh

n) = φ(

h

n)− φ(0) (34)

12

where φ is defined asφ(u) = Ft+ kh

n((zn)uei

t+ khn

−, vt, kh

n)

Since F is vertically differentiable, φ is differentiable and

φ′(u) = ∂iFt+ khn((zn)uei

t+ khn

−, vt, kh

n)

is right-continuous. Sinced∞((xt, vt), ((z

n)ueit+ kh

n−, vt, kh

n)) ≤ h,

φ′(u) > α hence:n∑

k=1

Ft+ khn(znt+ kh

n

, vt, khn)− Ft+ kh

n(znt+kh

n−, vt, kh

n) > αh.

On the other hand

Ft+ khn(znt+ kh

n−, vt, kh

n)− F

t+ (k−1)hn

(znt+

(k−1)hn

, vt, (k−1)h

n

) = ψ(h

n)− ψ(0)

whereψ(u) = F

t+ (k−1)h+u

n

(znt+ (k−1)h+u

n

, vt, (k−1)h+u

n

)

so that ψ is right-differentiable on ]0, hn [ with right-derivative:

ψ′r(u) = D

t+ (k−1)h+u

n

Ft+ (k−1)h+u

n

(znt+

(k−1)h+u

n

, vt, (k−1)h+u

n

)

Since F ∈ C0,0l ([0, T ]), ψ is also left-continuous by Lemma 2.5 so

n∑

k=1

Ft+ khn(znt+ kh

n−, vt,kh

n)− F

t+ (k−1)hn

(znt+

(k−1)hn

, vt, (k−1)h

n

) =

∫ h

0

Dt+uF (znt+u, vt,u)du

Noting that:

d∞((znt+u, vt,u), (zt+u, vt,u)) ≤h

nwe obtain that:

Dt+uF (znt+u, vt,u) →

n→+∞Dt+uF (zt+u, vt,u) = 0

since the path of zt+u is continuous. Moreover|DtFt+u(z

nt+u, vt,u)| ≤ 1 since d∞((znt+u, vt,u), (xt, vt) ≤ h, so by dominated convergence the integral

goes to 0 as n→ ∞. Writing:

Ft+h(z, vt,h)− Ft(x, v) = [Ft+h(z, vt,h)− Ft+h(zn, vt,h)] + [Ft+h(z

n, vt,h)− Ft(x, v)]

and taking the limit on n→ ∞ leads to Ft+h(z, vt,h)− Ft(x, v) ≥ αh, a contradiction.

The above result implies in particular that, if ∇xFi ∈ C1,1,D∇xF ∈ B, and F 1(x, v) = F 2(x, v)

for any continuous path x, then ∇2xF

1 and ∇2xF

2 must also coincide on continuous paths.We now show that the same result can be obtained under the weaker assumption that F i ∈ C1,2,

using a probabilistic argument. Interestingly, while the previous result on the uniqueness of thefirst vertical derivative is based on the fundamental theorem of calculus, the proof of the followingtheorem is based on its stochastic equivalent, the Ito formula [15, 16].

13

Theorem 3.8. If F 1, F 2 ∈ C1,2 such that DF 1,DF 2 ∈ B coincide on continuous paths:

∀(x, v) ∈ C0([0, T ],Rd)× ST , ∀t ∈ [0, T [, F 1

t (xt, vt) = F 2t (xt, vt) (35)

then their second vertical derivatives also coincide on continuous paths:

∀(x, v) ∈ C0([0, T ],Rd)× ST , ∀t ∈ [0, T [, ∇2

xF1t (xt, vt) = ∇2

xF2t (xt, vt)

Proof. Let F = F 1 − F 2. Assume now that there exists some (x, v) ∈ C0([0, T ],Rd)× ST such that

for some 1 ≤ i ≤ d and t0 ∈ [0, T [, and some direction h ∈ Rd, ‖h‖ = 1, th∇2xFt0(xt0 , vt0−).h > 0,

and denote α = 12th∇2

xFt0(xt0 , vt0−).h. We will show that this leads to a contradiction. We alreadyknow that ∇xFt(xt, vt) = 0 by theorem 3.7. Let η > 0 be small enough so that:

∀t′ ≤ t0, ∀(x′, v′) ∈ D([0, t′],Rd)× St′ , d∞((xt, vt), (x

′, v′)) < η ⇒

|Ft′(x′, v′)| < |Ft0(xt0 , vt0−)|+ 1, |∇xFt′(x

′, v′)| < 1, |Dt′F (x′, v′)| < 1, th∇2

xFt′(x′, v′).h > α (36)

Choose t < t0 such that d∞((xt, vt), (xt0 , vt0−)) <η2 and denote ǫ = η

2 ∧ (t0 − t). Let W be a one

dimensional Brownian motion on some probability space (Ω,B,P), (Bs) its natural filtration, andlet

τ = infs > 0, |W (s)| =ǫ

2 (37)

Define, for t′ ∈ [0, T ],

U(t′) = x(t′)1t′≤t + (x(t) +W ((t′ − t) ∧ τ)h)1t′>t (38)

and notice that for all s < ǫ2 ,

d∞((Ut+s, vt,s), (xt, vt)) < ǫ (39)

Define the following piecewise constant approximations of the stopped process W τ :

Wn(s) =

n−1∑

i=0

W (iǫ

2n∧ τ)1s∈[i ǫ

2n ,(i+1) ǫ2n ) +W (

ǫ

2∧ τ)1s= ǫ

2, 0 ≤ s ≤

ǫ

2n(40)

Denoting

Z(s) = Ft+s(Ut+s, vt,s), s ∈ [0, T − t] (41)

Un(t′) = x(t′)1t′≤t + (x(t) +Wn((t′ − t) ∧ τ)h)1t′>t Zn(s) = Ft+s(Unt+s, vt,s) (42)

we have the following decomposition:

Z(ǫ

2)− Z(0) = Z(

ǫ

2)− Zn(

ǫ

2) +

n∑

i=1

Zn(iǫ

2n)− Zn(i

ǫ

2n−)

+n−1∑

i=0

Zn((i+ 1)ǫ

2n−)− Zn(i

ǫ

2n) (43)

14

The first term in (43) goes to 0 almost surely since

d∞((Ut+ ǫ2, vt, ǫ

2), (Unt+ ǫ

2, vt, ǫ

2))n→∞→ 0. (44)

The second term in (43) may be expressed as

Zn(iǫ

2n)− Zn(i

ǫ

2n−) = φi(W (i

ǫ

2n)−W ((i− 1)

ǫ

2n))− φi(0) (45)

where:φi(u, ω) = Ft+i ǫ

2n(Un,uht+i ǫ

2n−(ω), vt,i ǫ2n)

Note that φi(u, ω) is measurable with respect to B(i−1)ǫ/2n whereas its argument in (45) is indepen-

dent with respect to B(i−1)ǫ/2n. Let Ω1 = ω ∈ Ω, t 7→ W (t, ω) continuous. Then P(Ω1) = 1and for any ω ∈ Ω1, φi(., ω) is C

2 with:

φ′i(u, ω) = ∇xFt+i ǫ2n(Un,uht+i ǫ

2n−(ω), vt,i ǫ2n)h

φ′′i (u, ω) =th∇2

xFt+i ǫ2n(Un,uht+i ǫ

2n−(ω), vt,i ǫ2n).h (46)

So, using the above arguments we can apply the Ito formula to (45) for each ω ∈ Ω1. We thereforeobtain, summing on i and denoting i(s) the index such that s ∈ [(i − 1) ǫ2n , i

ǫ2n ):

n∑

i=1

Zn(iǫ

2n)− Zn(i

ǫ

2n−) =

∫ ǫ2

0

∇xFt+i(s) ǫ2n(Un,uht+i(s) ǫ

2n−, vt,i(s) ǫ2n)hdW (s)

+

∫ ǫ2

0

th.∇2xFt+i(s) ǫ

2n(Un,uht+i(s) ǫ

2n−, vt,i(s) ǫ2n).hds (47)

Since the first derivative is bounded by (36), the stochastic integral is a martingale, so takingexpectation leads to:

E[

n∑

i=1

Zn(iǫ

2n)− Zn(i

ǫ

2n−)] > α

ǫ

2(48)

Now

Zn((i+ 1)ǫ

2n−)− Zn(i

ǫ

2n) = ψ(

ǫ

2n)− ψ(0) (49)

where

ψ(u) = Ft+(i−1) ǫ2n+u(U

nt+(i−1) ǫ

2n ,u, vt,(i−1) ǫ

2n+u) (50)

is right-differentiable with right derivative:

ψ′(u) = DtFt+(i−1) ǫ2n+u(U

n(i−1) ǫ

2n ,u, vt,(i−1) ǫ

2n+u) (51)

Since F ∈ C0,0l ([0, T ]), ψ is left-continuous by Lemma 2.5 and the fundamental theorem of calculus

yields:

n−1∑

i=0

Zn((i+ 1)ǫ

2n−)− Zn(i

ǫ

2n) =

∫ ǫ2

0

Dt+sF (Unt+(i(s)−1) ǫ

2n+u, vt,s)ds (52)

15

The integrand converges to DtFt+s(Ut+(i(s)−1) ǫ2n+u, vt,s) = 0 since DtF is zero whenever the first

argument is a continuous path. Since this term is also bounded, by dominated convergence theintegral converges almost surely to 0.It is obvious that Z( ǫ2 ) = 0 since F (x, v) = 0 whenever x is a continuous path. On the other hand,since all derivatives of F appearing in (43) are bounded, the dominated convergence theorem allowsto take expectations of both sides in (43) with respect to the Wiener measure and obtain α ǫ2 = 0, acontradiction.

4 Functional Ito calculus

4.1 Functional Ito formula

We are now ready to state a functional change of variable formula which extends the Ito formula topath-dependent functionals of a semimartingale:

Theorem 4.1 (Functional Ito formula). Let F ∈ C1,2b . For any t ∈ [0, T [,

Ft(Xt, At)− F0(X0, A0) =

∫ t

0

DuF (Xu, Au)du+

∫ t

0

∇xFu(Xu, Au).dX(u)

+

∫ t

0

1

2tr[t∇2

xFu(Xu, Au) d[X ](u)] a.s. (53)

In particular, for any F ∈ C1,2b , Y (t) = Ft(Xt, At) is a semimartingale.

We note that:

• the dependence of F on the second variable A does not enter the formula (53). Indeed, underthe assumption (7) variations in A lead to “higher order” terms which do not contribute.

• as expected from Theorems 3.7 and 3.8 in the case whereX is continuous then Y = F (X,A) de-pends on F and its derivatives only via their values on continuous paths. More precisely, Y canbe reconstructed from the second-order jet of F on C =

⋃

t∈[0,T [C0([0, t],Rd)×D([0, t], S+

d ) ⊂Υ.

The basic idea of the proof, as in the classical derivation of the Ito formula [8, 24, 28], is to approx-imate the path of X using piecewise constant predictable processes along a subdivision of [0, T ]. Acrucial remark, due to Dupire [10], is that the variations of a functional along a piecewise constantpath may be decomposed into successive “horizontal” and “vertical” increments, involving only thepartial functions used in the definitions of the pathwise derivatives (Definitions 3.1 and 3.2). Thisallows to express the functional F along a piecewise constant path in the form (53). The last stepis to take limits along a sequence of piecewise constant approximations of X , using the continu-ity properties of the pathwise derivatives. The control of the remainder terms is somewhat moreinvolved than in the usual proof of the Ito formula given that we are dealing with functionals.

Proof. Let us first assume that X does not exit a compact set K and that ‖A‖∞ ≤ R for someR > 0. Let us introduce a sequence of random subdivision of [0, t], indexed by n, as follows: starting

16

with the deterministic subdivision tni = it2n , i = 0..2n we add the time of jumps of A of size greater

or equal to 1n . We define the following sequence of stopping times:

τn0 = 0 τnk = infs > τnk−1|2ns ∈ N or |A(s)−A(s−)| >

1

n ∧ t (54)

The following arguments apply pathwise. Lemma A.3 ensures that ηn = sup|A(u) − A(τni )| +|X(u)−X(τni )|+

t2n , i ≤ 2n, u ∈ [τni , τ

ni+1) →n→∞ 0.

Denote nX =∑∞

i=0X(τni+1)1[τni,τn

i+1)+X(t)1t which is a non-adapted cadlag piecewise constant

approximation of Xt, and nA =∑∞i=0 A(τ

ni )1[τn

i,τn

i+1)+A(t)1t which is an adapted cadlag piecewise

constant approximation of At.Start with the decomposition:

Fτni+1

(nXτni+1−

,nAτni+1−

)− Fτni(nXτn

i−,nAτn

i−) = Fτn

i+1(nXτn

i+1−,nAτn

i,hn

i)− Fτn

i(nXτn

i,nAτn

i)

+ Fτni(nXτn

i,n Aτn

i−)− Fτn

i(nXτn

i−,nAτn

i−)(55)

where we have used the fact that F has predictable dependence in the second variable to haveFτn

i(nXτn

i,nAτn

i) = Fτn

i(nXτn

i,nAτn

i−). The first term in can be written ψ(hni )− ψ(0) where:

ψ(u) = Fτni+u(nXτn

i,u,nAτn

i,u) (56)

Since F ∈ C1,2([0, T ]), ψ is right-differentiable, and moreover by lemma 2.5, ψ is left-continuous, so:

Fτni+1

(nXτni,hn

i,nAτn

i,hn

i)− Fτn

i(nXτn

i,nAτn

i) =

∫ τni+1−τ

ni

0

Dτni+uF (nXτn

i,u,nAτn

i,u)du (57)

The second term in (55) can be written φ(X(τni+1)−X(τni ))−φ(0) where φ(u) = Fτni(nX

uτni−,nAτn

i).

Since F ∈ C1,2b , φ is a C2 functional parameterized by a Fτi-measurable random variable, and

φ′(u) = ∇xFτni(nX

uτni−,nAτn

i,hi

),φ′′(u) = ∇2xFτn

i(nX

uτni−,nAτn

i,hi

). Applying the Ito formula to φ

between times 0 and τi+1− τi and the (Fτi+s)s≥0) continuous semimartingale (X(τi+s))s≥0, yields:

φ(X(τni+1)−X(τni ) )− φ(0) =

∫ τni+1

τni

∇xFτni(nX

X(s)−X(τni )

τni− ,nAτn

i)dX(s)

+1

2

∫ τni+1

τni

tr[

t∇2xFτn

i(nX

X(s)−X(τni )

τni − ,nAτn

i)d[X ](s)

]

(58)

Summing over i = 0 to ∞ and denoting i(s) the index such that s ∈ [τni(s), τni(s)+1), we have

shown:

Ft(nXt,nAt)− F0(X0, A0) =

∫ t

0

DsF (nXτni(s)

,s−τni(s),nAτn

i(s),s−τn

i(s))ds

+

∫ t

0

∇xFτni(s)+1

(nXX(s)−X(τn

i(s))

τni(s)

− ,nAτni(s)

,hi(s))dX(s)

+1

2

∫ t

0

tr[

∇2xFτn

i(s)(nX

X(s)−X(τni(s))

τni(s)

− ,nAτni(s)

).d[X ](s)]

(59)

17

Ft(nXt,nAt) converges to Ft(Xt, At) almost surely. All the approximations of (X,A) appearing inthe various integrals have a d∞-distance from (Xs, As) less than ηn hence all the integrands ap-pearing in the above integrals converge respectively to DsF (Xs, As),∇xFs(Xs, As),∇

2xFs(Xs, As)

as n → ∞ by d∞ fixed time continuity for DF and d∞ left-continuity for the vertical derivatives.Since the derivatives are in B the integrands in the various above integrals are bounded by a constantdependant only on F ,K and R and t hence does not depend on s nor on ω. The dominated conver-gence and the dominated convergence theorem for the stochastic integrals [28, Ch.IV Theorem 32]then ensure that the Lebesgue-Stieltjes integrals converge almost surely, and the stochastic integralin probability, to the terms appearing in (53) as n→ ∞.

Consider now the general case where X and A may be unbounded. Let Kn be an increasingsequence of compact sets with

⋃

n≥0Kn = Rd and denote

τn = infs < t|Xs /∈ Kn or |As| > n ∧ t

which are optional times. Applying the previous result to the stopped process (Xt∧τn , At∧τn) leadsto:

Ft(Xt∧τn , At∧τn)− F0(Z0, A0) =

∫ t∧τn

0

DuFu(Xu, Au)du

+1

2

∫ t∧τn

0

tr(

t∇2xFu(Xu, Au)d[X ](u)

)

+

∫ t∧τn

0

∇xFu(Xu, Au).dX +

∫ t

t∧τn

DuF (Xu∧τn , Au∧τn)du (60)

The terms in the first line converges almost surely to the integral up to time t since t∧τn = t almostsurely for n sufficiently large. For the same reason the last term converges almost surely to 0.

Remark 4.2. The above proof is probabilistic and makes use of the Ito formula (for functions ofsemimartingales). In the companion paper [5] we give a alternative, non-probabilistic, proof ofTheorem 4.1, using the analytical approach of Follmer [12], which allows X to have discontinuous(cadlag) trajectories.

Example 10. If Ft(xt, vt) = f(t, x(t)) where f ∈ C1,2([0, T ]× Rd), (53) reduces to the standard Ito

formula.

Example 11. For integral functionals of the form

Ft(xt, vt) =

∫ t

0

g(x(u))v(u)du (61)

where g ∈ C0(Rd), the Ito formula reduces to the trivial relation

Ft(Xt, At) =

∫ t

0

g(X(u))A(u)du (62)

since the vertical derivatives are zero in this case.

18

Example 12. For a scalar semimartingale X , applying the formula to Ft(xt, vt) = x(t)2 −∫ t

0v(u)du

yields the well-known Ito product formula:

X(t)2 − [X ](t) =

∫ t

0

2X.dX (63)

Example 13. For the Doleans functional (Ex. 4)

Ft(xt, vt) = ex(t)−12

∫t

0v(u)du (64)

the formula (53) yields the well-known integral representation

exp(X(t)−1

2[X ](t) ) =

∫ t

0

eX(u)− 12 [X](u)dX(u) (65)

An immediate corollary of Theorem 4.1 is that, if X is a local martingale, any C1,2b functional of

X which has finite variation is equal to the integral of its horizontal derivative:

Corollary 4.3. If X is a local martingale and F ∈ C1,2b , the process Y (t) = Ft(Xt, At) has finite

variation if only if ∇xFt(Xt, At) = 0 d[X ]× dP-almost everywhere.

Proof. Y (t) is a continuous semimartingale by Theorem 4.1, with canonical decomposition given by(53). If Y has finite variation, then by formula (53), its continuous martingale component should

be zero i.e.∫ t

0 ∇xFt(Xt, At).dX(t) = 0 a.s. Computing the quadratic variation of this martingalewe obtain

∫ T

0

tr(

t∇xFt(Xt, At).∇xFt(Xt, At).d[X ])

= 0

which implies in particular that ‖∂iFt(Xt, At)‖2 = 0 d[X i] × dP-almost everywhere for i = 1..d.

Thus, ∇xFt(Xt, At) = 0 for (t, ω) /∈ A ⊂ [0, T ]× Ω where∫

A d[Xi]× dP = 0 for i = 1..d.

4.2 Intrinsic nature of the vertical derivative

Whereas the functional representation (29) of a (Ft−adapted) process Y is not unique, Theorem4.1 implies that the process ∇xFt(Xt, At) has an intrinsic character i.e. independent of the chosenrepresentation:

Corollary 4.4. Let F 1, F 2 be two functionals in C1,2b , such that:

∀t ∈ [0, T [, F 1t (Xt, At) = F 2

t (Xt, At) P− a.s. (66)

Then, outside an evanescent set:

t[∇xF1t (Xt, At)−∇xF

2t (Xt, At)]A(t−)[∇xF

1t (Xt, At)−∇xF

2t (Xt, At)] = 0 (67)

Proof. Let X(t) = B(t) +M(t) where B is a continuous process with finite variation and M is acontinuous local martingale. There exists Ω1 ⊂ Ω such that P(Ω1) = 1 and for ω ∈ Ω the path oft 7→ X(t, ω) is continuous and t 7→ A(t, ω) is cadlag. Theorem 4.1 implies that the local martingalepart of the null process 0 = F 1(Xt, At)− F 2(Xt, At) can be written:

0 =

∫ t

0

[

∇xF1u (Xu, Au)−∇xF

2u (Xu, Au)

]

dM(u) (68)

19

Considering its quadratic variation, we have, on Ω1

0 =

∫ t

0

1

2t[∇xF

1u (Xu, Au)−∇xF

2u (Xu, Au)]A(u−)[∇xF

1u (Xu, Au)−∇xF

2u (Xu, Au)]du (69)

Noticing that on Ω1 the integrand in (69) is left-continuous by Lemma 2.5 (∇xF1(Xt, At) =

∇xF1(Xt−, At− because X is continuous and F is slow in the second variable), this yields that,

for all t < T and ω ∈ Ω1,

t[∇xF1u (Xu, Au)−∇xF

2u (Xu, Au)]A(u−)[∇xF

1u (Xu, Au)−∇xF

2u (Xu, Au) = 0

In the case where for all t < T , A(t−) is almost surely positive definite, Corollary 4.4 allows todefine intrinsically the pathwise derivative of a process Y which admits a functional representationY (t) = Ft(Xt, At).

Definition 4.5 (Vertical derivative of a process). Define C1,2b (X) the set of Ft-adapted processes Y

which admit a functional representation in C1,2b :

C1,2b (X) = Y, ∃F ∈ C

1,2b Y (t) = Ft(Xt, At) P− a.s. (70)

If A(t−) is almost-surely non-singular then for any Y ∈ C1,2b (X), the predictable process:

∇XY (t) = ∇xFt(Xt, At)

is uniquely defined up to an evanescent set, independently of the choice of F ∈ C1,2b in the represen-

tation (29). We will call ∇XY the vertical derivative of Y with respect to X .

In particular this construction applies to the case where X is a standard Brownian motion, whereA = Id, so we obtain the existence of a vertical derivative process for C1,2

b Brownian functionals:

Definition 4.6 (Vertical derivative of non-anticipative Brownian functionals). LetW be a standardd-dimensional Brownian motion. For any Y ∈ C1,2

b (W ) with representation Y (t) = Ft(Wt, t), thepredictable process

∇WY (t) = ∇xFt(Wt, t)

is uniquely defined up to an evanescent set, independently of the choice of F ∈ C1,2b .

5 Stochastic integral representation of martingales

We consider now the case where X is a continuous martingale. The functional Ito formula (Theorem4.1) then leads to an explicit martingale representation formula for Ft-martingales in C1,2

b (X). Thisresult may be seen as a non-anticipative counterpart of the Clark-Haussmann-Ocone formula [4, 26,14] and generalizes explicit martingale representation formulas previously obtained in a Markoviancontext by Elliott and Kohlmann [11] and Jacod et al. [17].

20

5.1 A martingale representation formula

Consider an FT measurable random variable H with E|H | <∞ and consider the martingale Y (t) =E[H |Ft]. If Y admits a representation Y (t) = Ft(Xt, At) where F ∈ C

1,2b , we obtain the following

stochastic integral representation for Y in terms of its derivative ∇XY with respect to X :

Theorem 5.1. If Y (t) = Ft(Xt, At) for some functional F ∈ C1,2b , then:

Y (T ) = E[Y (T )] +

∫ T

0

∇xFt(Xt, At)dX(t) (71)

Note that regularity assumptions are given not on H = Y (T ) but on the functionals Y (t) =E[H |Ft], t < T , which is typically more regular than H itself.

Proof. Theorem 4.1 implies that for t ∈ [0, T [:

Y (t) = [

∫ t

0

DuF (Xu, Au)du +1

2

∫ t

0

tr[t∇2xFu(Xu, Au)d[X ](u)]

+

∫ t

0

∇xFu(Xu, Au)dX(u) (72)

Given the regularity assumptions on F , the first term in this sum is a finite variation process while thesecond is a local martingale. However, Y is a martingale and the decomposition of a semimartingaleas sum of finite variation process and local martingale is unique. Hence the first term is 0 and:Y (t) =

∫ t

0Fu(Xu, Au)dXu. Since F ∈ C

0,0l ([0, T ]) Y (t) has limit FT (XT , AT ) as t → T , so the

stochastic integral also converges.

Example 14.

If the Doleans-Dade exponential eX(t)− 12 [X](t) is a martingale, applying Theorem 5.1 to the functional

Ft(xt, vt) = ex(t)−∫

t

0v(u)du yields the familiar formula:

eX(t)− 12 [X](t) = 1+

∫ t

0

eX(s)− 12 [X](s)dX(s) (73)

If X(t)2 is integrable, applying Theorem 5.1 to the functional Ft(x(t), v(t)) = x(t)2 −∫ t

0v(u)du, we

obtain the well-known Ito product formula

X(t)2 − [X ](t) =

∫ t

0

2X(s)dX(s) (74)

5.2 Relation with the Malliavin derivative

The reader familiar with Malliavin calculus is by now probably intrigued by the relation betweenthe pathwise calculus introduced above and the stochastic calculus of variations as introduced byMalliavin [23] and developed by Bismut [2, 3], Stroock [30], Shigekawa [29], Watanabe [33] andothers.

To investigate this relation, consider the case where X(t) = W (t) is the Brownian motion andP the Wiener measure. Denote by Ω0 the canonical Wiener space (C0([0, T ],R

d), ‖.‖∞,P) endowedwith its Borelian σ-algebra, the filtration of the canonical process.

21

Consider an FT -measurable functional H = H(X(t), t ∈ [0, T ]) = H(XT ) with E[|H |2] < ∞and define the martingale Y (t) = E[H |Ft]. If H is differentiable in the Malliavin sense [23, 25, 30]e.g. H ∈ D1,2 with Malliavin derivative DtH , then the Clark-Haussmann-Ocone formula [18, 26, 25]gives a stochastic integral representation of the martingale Y in terms of the Malliavin derivative ofH :

H = E[H ] +

∫ T

0

pE[DtH |Ft]dWt (75)

where pE[DtH |Ft] denotes the predictable projection of the Malliavin derivative. Similar represen-tations have been obtained under a variety of conditions [2, 7, 11, 1].

However, as shown by Pardoux and Peng [27, Prop. 2.2] in the Markovian case, one does notreally need the full specification of the (anticipative) process (DtH)t∈[0,T ] in order to recover the(predictable) martingale representation of H . Indeed, when X is a (Markovian) diffusion process,Pardoux & Peng [27, Prop. 2.2] show that in fact the integrand is given by the “diagonal” Malliavinderivative DtYt, which is non-anticipative.

Theorem 5.1 shows that this result holds beyond the Markovian case and yields an explicitnon-anticipative representation for the martingale Y as a pathwise derivative of the martingale Y ,provided that Y ∈ C1,2(X).

The uniqueness of the integrand in the martingale representation (71) leads to (with a slightabuse of notation):

pE[DtH |Ft] = ∇W (E[H |Ft]) , dt× dP− a.s. (76)

Theorem 5.2. Denote by

• P the set of Ft-adapted processes on [0, T ] with values in L1(Ω,FT ,P).

• Lp([0, T ]× Ω) the set of (anticipative) processes φ on [0, T ] with E∫ T

0‖φ(t)‖pdt <∞.

• D the Malliavin derivative operator, which associates to a random variable H ∈ D1,1(0, T ) the(anticipative) process (DtH)t∈[0,T ] ∈ L1([0, T ]× Ω).

• H the set of Malliavin-differentiable functionals H ∈ D1,1(0, T ) whose predictable projectionHt =

pE[H |Ft] admits a C1,2b (W ) version:

H = H ∈ D1,1, ∃Y ∈ C1,2b (W ), E[H |Ft] = Y (t) dt× dP− a.e

Then the following diagram is commutative, in the sense of dt× dP almost everywhere equality:

HD→ L1([0, T ]× Ω)

↓(pE[.|Ft])t∈[0,T ] ↓(pE[.|Ft])t∈[0,T ]

C1,2b (W )

∇W→ P

Proof. The Clark-Haussmann-Ocone formula extended to D1,1 in [18] gives

H = E[H ] +

∫ T

0

pE[DtH |Ft]dWt (77)

22

where pE[DtH |Ft] denotes the predictable projection of the Malliavin derivative. On other handtheorem 5.1 gives:

H = E[H ] +

∫ T

0

∇WE[H |Ft]dW (t) (78)

Hence: pE[DtH |Ft] = ∇WE[H |Ft], dt× dP almost everywhere.

From a computational viewpoint, unlike the Clark-Haussmann-Ocone representation which re-quires to simulate the anticipative process DtH and compute conditional expectations, ∇XY onlyinvolves non-anticipative quantities which can be computed in a pathwise manner. This implies theusefulness of (71) for the numerical computation of martingale representations, a topic which wefurther explore in a forthcoming work.

6 Weak derivatives and integration by parts for stochastic

integrals

Assume now thatX is a continuous, square-integrable real-valued martingale. We will now show that∇X may be extended to a weak derivative which acts as the inverse of the Ito stochastic integrals,that is, an operator which verifies

∇X

(∫

φ.dX

)

= φ, dt× dP− a.s. (79)

for square-integrable stochastic integrals of the form:

Y (t) =

∫ t

0

φsdX(s) where E

[∫ t

0

φ2sd[X ](s)

]

<∞ (80)

Remark 6.1. The construction in this section does not require the assumption of absolute continuityfor [X ], since the functionals used to prove lemma 6.6 do not depend on A.

Let L2(X) be the Hilbert space of progressively-measurable processes φ such that:

||φ||2L2(X) = E

[∫ t

0

φ2sd[X ](s)

]

<∞ (81)

and I2(X) be the space of square-integrable stochastic integrals with respect to X :

I2(X) =

∫ .

0

φ(t)dX(t), φ ∈ L2(X) (82)

endowed with the norm

||Y ||22 = E[Y (T )2] (83)

The Ito integral φ 7→∫ .

0 φsdX(s) is then a bijective isometry from L2(X) to I2(X) [28].

23

Definition 6.2 (Space of test processes). The space of test processes D(X) is defined as

D(X) = C1,2b (X) ∩ I2(X) (84)

Theorem 5.1 allows to define intrinsically the vertical derivative of a process in D(X) as anelement of L2(X).

Definition 6.3. Let Y ∈ D(X), define the process ∇XY ∈ L2(X) as the equivalence classof ∇xFt(Xt, At), which does not depend on the choice of the representation functional Y (t) =Ft(Xt, At)

Theorem 6.4 (Integration by parts on D(X)). Let Y, Z ∈ D(X). Then:

E [Y (T )Z(T )] = E

[

∫ T

0

∇XY (t)∇XZ(t)d[X ](t)

]

(85)

Proof. Let Y, Z ∈ D(X) ⊂ C1,2b (X). Then Y, Z are martingales with Y (0) = Z(0) = 0 and

E[|Y (T )|2] <∞, E[|Z(T )|2] <∞. Applying Theorem 5.1 to Y and Z, we obtain

E [Y (T )Z(T )] = E[

∫ T

0

∇XY dX

∫ T

0

∇XZdX ]

Applying the Ito isometry formula yields the result.

Using this result, we can extend the operator ∇X in a weak sense to a suitable space of thespace of (square-integrable) stochastic integrals, where ∇XY is characterized by (85) being satisfiedagainst all test processes.

The following definition introduces the Hilbert space W1,2(X) of martingales on which ∇X actsas a weak derivative, characterized by integration-by-part formula (85). This definition may be alsoviewed as a non-anticipative counterpart of Wiener-Sobolev spaces in the Malliavin calculus [23, 29].

Definition 6.5 (Martingale Sobolev space). The Martingale Sobolev space W1,2(X) is defined asthe closure in I2(X) of D(X).

The Martingale Sobolev space W1,2(X) is in fact none other than I2(X), the set of square-integrable stochastic integrals:

Lemma 6.6. ∇XY, Y ∈ D(X) is dense in L2(X) and

W1,2(X) = I2(X).

Proof. We first observe that the set U of “cylindrical” processes of the form

φn,f,(t1,..,tn)(t) = f(X(t1), ..., X(tn))1t>tn

where n ≥ 1, 0 ≤ t1 < .. < tn ≤ T and f ∈ C∞b (Rn,R) is a total set in L2(X) i.e. the linear span of

U is dense in L2(X).

24

For such an integrand φn,f,(t1,..,tn), the stochastic integral with respect to X is given by themartingale

Y (t) = IX(φn,f,(t1,..,tn))(t) = Ft(Xt, At)

where the functional F is defined on Υ as:

Ft(xt, vt) = f(x(t1−), ..., x(tn−))(x(t) − x(tn))1t>tn ∈ C0,0l

so that:∇xFt(xt, vt) = f(xt1−, ..., xtn−)1t>tn ∈ C

0,0l

∇2xFt(xt, vt) = 0,DtF (xt, vt) = 0

which prove that F ∈ C1,2b . Hence, Y ∈ C1,2

b (X). Since f is bounded, Y is obviously squareintegrable so Y ∈ D(X). Hence IX(U) ⊂ D(X).

Since IX is a bijective isometry from L2(X) to I2(X), the density of U in L2(X) entails thedensity of IX(U) in I2(X), so W 1,2(X) = I2(X).

Theorem 6.7 (Weak derivative on W1,2(X)). The vertical derivative ∇X : D(X) 7→ L2(X) isclosable on W1,2(X). Its closure defines a bijective isometry

∇X : W1,2(X) 7→ L2(X)∫ .

0

φ.dX 7→ φ (86)

characterized by the following integration by parts formula: for Y ∈ W1,2(X), ∇XY is the uniqueelement of L2(X) such that

∀Z ∈ D(X), E[Y (T )Z(T )] = E

[

∫ T

0

∇XY (t)∇XZ(t)d[X ](t)

]

. (87)

In particular, ∇X is the adjoint of the Ito stochastic integral

IX : L2(X) 7→ W1,2(X)

φ 7→

∫ .

0

φ.dX (88)

in the following sense:

∀φ ∈ L2(X), ∀Y ∈ W1,2(X), < Y, IX(φ) >W1,2(X)=< ∇XY, φ >L2(X) (89)

i.e. E[Y (T )

∫ T

0

φ.dX ] = E[

∫ T

0

∇XY φ.d[X ] ] (90)

Proof. Any Y ∈ W1,2(X) may be written as Y (t) =∫ t

0φ(s)dX(s) for some φ ∈ L2(X), which is

uniquely defined d[X ]×dP a.e. The Ito isometry formula then guarantees that (87) holds for φ. Onestill needs to prove that (87) uniquely characterizes φ. If some process ψ also satisfies (87), then,denoting Y ′ = IX(ψ) its stochastic integral with respect to X , (87) then implies that U = Y ′ − Yverifies

∀Z ∈ D(X), < U, Z >W1,2(X)= E[U(T )Z(T )] = 0

25

which implies U = 0 d[X ] × dP a.e. since by construction D(X) is dense in W1,2(X). Hence,∇X : D(X) 7→ L2(X) is closable on W1,2(X)

This construction shows that ∇X : W1,2(X) 7→ L2(X) is a bijective isometry which coincideswith the adjoint of the Ito integral on W1,2(X).

Thus, Ito’s stochastic integral IX with respect to X , viewed as the map

IX : L2(X) 7→ W1,2(X)

admits an inverse on W1,2(X) which is a weak form of the vertical derivative ∇X introduced inDefinition 3.2.

Remark 6.8. In other words, we have established that for any φ ∈ L2(X) the relation

∇X (φ.X) (t) = φ(t) where (φ.X)(t) =

∫ t

0

φ(u)dX(u) (91)

holds in a weak sense.

In particular these results hold when X =W is a Brownian motion. We can now restate a square-integrable version of theorem 5.2, which holds on D1,2, and where the operator ∇W is defined inthe weak sense of theorem 6.7.

Theorem 6.9 (Lifting theorem). Consider Ω0 = C0([0, T ],Rd) endowed with its Borelian σ-algebra,

the filtration of the canonical process and the Wiener measure P. Then the following diagram iscommutative is the sense of dt× dP equality:

I2(W )∇W→ L2(W )

↑(E[.|Ft])t∈[0,T ] ↑(E[.|Ft])t∈[0,T ]

D1,2 D→ Lp([0, T ]× Ω)

Remark 6.10. With a slight abuse of notation, the above result can be also written as

∀H ∈ L2(Ω0,FT ,P), ∇W (E[H |Ft]) = E[DtH |Ft] (92)

In other words, the conditional expectation operator intertwines ∇W with the Malliavin derivative.

Thus, the conditional expectation operator (more precisely: the predictable projection on Ft)can be viewed as a morphism which “lifts” relations obtained in the framework of Malliavin calculusinto relations between non-anticipative quantities, where the Malliavin derivative and the Skorokhodintegral are replaced by the weak derivative operator ∇W and the Ito stochastic integral. Obviously,making this last statement precise is a whole research program, beyond the scope of this paper.

7 Functional equations for martingales

Consider now a semimartingale X whose characteristics are left-continuous functionals:

dX(t) = bt(Xt, At)dt+ σt(Xt, At)dW (t) (93)

where b, σ are non-anticipative functionals on Υ (in the sense of Definition 2.1) with values inRd-valued (resp. Rd×n, whose coordinates are in C

0,0l . The topological support in (C0([0, T ],R

d) ×

26

ST , ‖.‖∞) of the law of (X,A) is defined as the subset supp(X,A) of all paths (x, v) ∈ C0([0, T ],Rd)×

ST for which every neighborhood has positive measure:

supp(X,A) = (x, v) ∈ C0([0, T ],Rd)× ST | for any Borel neighborhood V of (x, v),P((X,A) ∈ V ) > 0

Functionals of X which have the (local) martingale property play an important role in controltheory and harmonic analysis. The following result characterizes a functional F ∈ C

1,2b which define

a local martingale as the solution to a functional version of the Kolmogorov backward equation:

Theorem 7.1 (Functional equation for C1,2 martingales). If F ∈ C1,2b and DF ∈ C

0,0l , then Y (t) =

Ft(Xt, At) is a local martingale if and only if F satisfies the functional partial differential equation:

DtF (xt, vt) + bt(xt, vt)∇xFt(xt, vt) +1

2tr[∇2

xF (xt, vt)σttσt(xt, vt)] = 0, (94)

on the topological support of (X,A) in (C0([0, T ],Rd)× ST , ‖.‖∞).

Proof. If F ∈ C1,2b , then applying Theorem 4.1 to Y (t) = Ft(Xt, At), (94) implies that the finite

variation term in (53) is almost-surely zero: Y (t) =∫ t

0 ∇xFt(Xt, At)dX(t), and also Y is continuousup to time T by left-continuity of F . Hence Y is a local martingale.

Conversely, assume that Y is a local martingale. Note that Y is left-continuous by Lemma 2.5.Suppose the functional relation (94) is not satisfied at some (x, v) belongs to the supp(X,A) ⊂C0([0, T ],R

d)× ST . Then there exists t0 < T , η > 0 and ǫ > 0 such that

|DtF (xt, vt) + bt(xt, vt)∇xFt(xt, vt) +1

2tr[∇2

xF (xt, vt)σttσt(xt, vt)]| > ǫ (95)

for t ∈ [t0 − η, t0], by left-continuity of the expression. By left-continuity of the expression for thed∞ norm, there exist an open neighborhood of (x, v) in C0([0, T ],R

d)×ST such that, for all (x′, v′)in this neighborhood and all t ∈ [t0 − η, t0]:

|DtF (x′t, v

′t) + bt(x

′t, v

′t)∇xFt(xt, vt) +

1

2tr[∇2

xF (x′t, v

′t)σt

tσt(x′t, v

′t)]| >

ǫ

2(96)

Since (X,A) belongs to this neighborhood with non-zero probability, it proves that:

DtF (Xt, At) + bt(Xt, At)∇xFt(xt, vt) +1

2tr[∇2

xF (Xt, At)σttσt(Xt, At)]| >

ǫ

2(97)

with non-zero dt×dP measure. Applying theorem 4.1 to the process Y (t) = Ft(Xt, At) then leads toa contradiction, because as a continuous local martingale its finite variation part should be null.

The martingale property of F (X,A) implies no restriction on the behavior of F outside supp(X,A)so one cannot hope for uniqueness of F on Υ in general. However, the following result gives a con-dition for uniqueness of a solution of (94) on supp(X,A):

Theorem 7.2 (Uniqueness of solutions). Let h be a continuous functional on (C0([0, T ])×ST , ‖.‖∞).Any solution F ∈ C

1,2b of the functional equation (94), verifying

FT (x, v) = h(x, v) (98)

E[ supt∈[0,T ]

|Ft(Xt, At)|] <∞ (99)

27

is uniquely defined on the topological support supp(X,A) of (X,A) in (C0([0, T ],Rd) × ST , ‖.‖): if

F 1, F 2 ∈ C1,2b ([0, T ]) verify (94)-(98)-(99) then

∀(x, v) ∈ supp(X,A), ∀t ∈ [0, T ] F 1t (xt, vt) = F 2

t (xt, vt). (100)

Proof. Let F 1 and F 2 be two such solutions. Theorem 7.1 shows that they are local martingales.The integrability condition (99) guarantees that they are true martingales, so that we have theequality: F 1

t (Xt, At) = F 2t (Xt, At) = E[h(XT , AT )|Ft] almost surely. Hence reasoning along the

lines of the proof of theorem 7.1 shows that F 1t (xt, vt) = F 2

t (xt, vt) if (x, v) ∈ supp(X,A).

Example 15. Consider a scalar diffusion

dX(t) = b(t,X(t))dt+ σ(t,X(t))dW (t) X(0) = x0 (101)

whose law Px0 is defined as the solution of the martingale problem [32] for the operator

Ltf =1

2σ2(t, x)∂2xf(t, x) + b(t, x)∂xf(t, x)

where b and σ are continuous and bounded functions, with σ bounded away from zero. We areinterested in computing the martingale

Y (t) = E[

∫ T

0

g(t,X(t))d[X ](t)|Ft] (102)

for a continuous bounded function g. The topological support of the process (X,A) under Px0 isthen given by the Stroock-Varadhan support theorem [31, Theorem 3.1.] which yields:

(x, (σ2(t, x(t)))t∈[0,T ]) |x ∈ C0(Rd, [0, T ]), x(0) = x0, (103)

From theorem 7.1 a necessary condition for Y to have a a functional representation Y = F (X,A)with F ∈ C

1,2b is that F verifies

DtF (xt, (σ2(u, x(u)))u≤t) + b(t, x(t))∇xFt(xt, (σ

2(u, x(u)))u∈[0,t]) (104)

+1

2σ2(t, x(t))∇2

xFt(xt, (σ2(u, x(u)))u∈[0,t]) = 0

together with the terminal condition:

FT (xT , (σ2(u, x(u))u∈[0,T ]) =

∫ T

0

g(t, x(t))σ2(t, x(t))dt (105)

for all x ∈ C0(Rd), x(0) = x0. Moreover, from theorem 7.2, we know that there any solution

satisfying the integrability condition:

E[ supt∈[0,T ]

|Ft(Xt, At)|] <∞ (106)

is unique on supp(X,A). If such a solution exists, then the martingale Ft(Xt, At) is a version of Y .To find such a solution, we look for a functional of the form:

28

Ft(xt, vt) =

∫ t

0

g(u, x(u))v(u)du+ f(t, x(t))

where f is a smooth C1,2 function. Elementary computation show that F ∈ C1,2b ; so F is solution of

the functional equation (104) if and only if f satisfies the Partial Differential Equation with sourceterm:

1

2σ2(t, x)∂2xf(t, x) + b(t, x)∂xf(t, x) + ∂tf(t, x) = −g(t, x)σ2(t, x) (107)

with terminal condition f(T, x) = 0

The existence of a solution f with at most exponential growth is then guaranteed by standard resultson parabolic PDEs [19]. In particular, theorem 7.2 guarantees that there is at most one solutionsuch that:

E[ supt∈[0,T ]

|f(t,X(t))|] <∞ (108)

Hence the martingale Y in (102) is given by

Y (t) =

∫ t

0

g(u,X(u))d[X ](u) + f(t,X(t))

where f is the unique solution of the PDE (107).

References

[1] H. Ahn, Semimartingale integral representation, Ann. Probab., 25 (1997), pp. 997–1010.

[2] J.-M. Bismut, A generalized formula of Ito and some other properties of stochastic flows, Z.Wahrsch. Verw. Gebiete, 55 (1981), pp. 331–350.

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A Proof of Theorem 2.6

In order to prove theorem 2.6 in the general case where A is just required to be cadlag, we need thefollowing three lemmas:

Lemma A.1. Let f be a cadlag function on [0, T ] and define ∆f(t) = f(t)− f(t−). Then

∇ǫ > 0, ∃η > 0, |x− y| ≤ η ⇒ |f(x)− f(y)| ≤ ǫ+ supt∈[x,y]

|∆f(t)| (109)

Proof. Assume the conclusion does not hold. Then there exists a sequence (xn, yn)n≥1 such thatxn ≤ yn, yn − xn → 0 but |f(xn)− f(yn)| > ǫ+ supt∈[xn,yn]|∆f(t)|. We can extract a convergentsubsequence (xψ(n)) such that xψ(n) → x. Noting that either an infinity of terms of the sequence areless than x or an infinity are more than x, we can extract monotone subsequences (un, vn)n≥1 whichconverge to x. If (un), (vn) both converge to x from above or from below, |f(un)−f(vn)| → 0 whichyields a contradiction. If one converges from above and the other from below, supt∈[un,vn]|∆f(t)| >|∆f(x)| but |f(un) − f(vn)| → |∆f(x)|, which results in a contradiction as well. Therefore (109)must hold.

Lemma A.2. If α ∈ R and V is an adapted cadlag process defined on a filtered probability space(Ω,F, (Ft)t≥0,P) and σ is a optional time, then:

τ = inft > σ, |V (t)− V (t−)| > α (110)

is a stopping time.

31

Proof. We can write that:

τ ≤ t =⋃

q∈Q⋂[0,t)

(σ ≤ t− q⋂

supt∈(t−q,t]

|V (u)− V (u−)| > α (111)

and

supu∈(t−q,t]

|V (u)− V (u−)| > α =⋃

n0>1

⋂

n>n0

sup1≤i≤2n

|V (t− qi− 1

2n)− V (t− q

i

2n)| > α (112)

thanks to the lemma A.1.

The following lemma is a consequence of lemma A.1:

Lemma A.3 (Uniform approximation of cadlag functions by step functions).Let h be a cadlag function on [0, T ] and (tnk )n≥0,k=0..n is a sequence of subdivisions 0 = tn0 < t1 <... < tnkn = T of [0, T ] such that:

sup0≤i≤k−1

|tni+1 − tni |n→∞→ 0 sup

u∈[0,T ]\tn0 ,...,tnkn

|∆f(u)|n→∞→ 0

then

supu∈[0,T ]

|h(u)−

kn−1∑

i=0

h(ti)1[tni ,tni+1)(u) + h(tnkn)1tnkn(u)|

n→∞→ 0 (113)

We can now prove the A cadlag case of Theorem 2.6, using lemma A.1.Proof of Theorem 2.6: Let us first prove that Ft(Xt, At) is adapted. Define the following

sequence of stopping times:

τN0 = 0 τNk = inft > τNk−1|2N t ∈ N or |v(t)− v(t−)| >

1

N ∧ t (114)

From lemma A.2, those functionals are stopping times. We define the stepwise approximations ofXt and At along the subdivision of index N :

XN (s) =

∞∑

k=0

XτNk1[τN

k,τN

k+1[(s) +X(t)1t(s)

AN (s) =∞∑

k=0

AτNk1[τN

k,τN

k+1)(t) +A(t)1t(s) (115)

as well as their truncations of rank K:

KXN(s) =

K∑

k=0

XτNk1[τN

k,τN

k+1)(s)

KAN (t) =

K∑

k=0

AτNk1[τN

k,τN

k+1)(t) (116)

32

First notice that:Ft(X

Nt , A

Nt ) = lim

K→∞Ft(KX

Nt ,K A

Nt ) (117)

because (KXNt ,K A

Nt ) coincides with (XN

t , ANt ) for K sufficiently large. The truncations

Fnt (KXNt ,K A

Nt )

are Ft-measurable as they are continuous functions of the random variables:

(X(τNk )1τNk

≤t, A(τNk )1τN

k≤t), k ≤ K

so their limit Ft(XNt , A

Nt ) is also Ft-measurable. Thanks to lemma A.3, XN

t and ANt convergeuniformly to Xt and At, hence Ft(X

Nt , A

Nt ) converges to Ft(Xt, At) since F is continuous at fixed

time.

Now to show optionality of Z(t) in point 2., we will exhibit it as limit of right-continuous adapted

processes. For t ∈ [0, T ], define in(t) to be the integer such that t ∈ [ iTn ,(i+1)Tn ). Define the process:

Znt = F (in(t))Tn

(X (in(t))Tn

, A (in(t))Tn

), which is piecewise-constant and has right-continuous trajectories,

and is also adapted by the first part of the theorem. Since F ∈ C0,0l , Zn(t) → Z(t) almost surely,

which proves that Z is optional.Point 3. follows from point 1. and lemma 2.5, since in both cases Ft(Xt, At) = Ft(Xt−, At−) henceZ has left-continuous trajectories.

33