a non-anticipative calculus for functionals of semimartingales · functional representation of...

Functional representation of non-anticipative processesPathwise derivatives of functionals

A pathwise change of variable formulaFunctional Ito formula

Martingale representation formulaWeak derivative and integration by parts formula

Functional equations for martingales

A non-anticipative calculusfor functionals of semimartingales

Rama Cont & David Fournie

Laboratoire de Probabilites et Modeles AleatoiresCNRS - Universite de Paris VI-VII

andColumbia University, New York

LEVY PROCESSES 2010

Rama Cont & David Fournie Functional Ito calculus





References

R. Cont & D Fournie (2010) A functional extension of the Itoformula, Comptes Rendus Acad. Sci., Vol. 348, 57–61.

R. Cont & D Fournie (2009) Functional Ito formula andstochastic integral representation of martingales,arxiv/math.PR.

R. Cont & D Fournie (2010) Change of variable formulas fornon-anticipative functionals on path space, Journal ofFunctional Analysis, Vol 259, 1043–1072.

R. Cont & D Fournie (2009) Calcul d’Ito fonctionnel sansprobabilites, Working paper.






Framework

Consider a cadlag ℝd -valued Ito-Levy semimartingale on(Ω,ℬ,ℬt ,ℙ):

X (t) =

∫ t

0�(u)du +

∫ t

0�(u).dWu +

∫ t

0

∫ℝd

z JX (du dz)

� integrable, � square integrable ℬt-adapted processes,JX (dtdz ;!) = JX (dtdz ;!)− �(dt, dz ;!) compensated jumpmeasure

Quadratic variation process [X ](t)

D([0,T ],ℝd) space of cadlag functions.

ℱt = ℱXt : natural filtration / history of X

C0([0,T ],ℝd) space of continuous paths.






Functional notation

For a path ! ∈ D([0,T ],ℝd), denote by

!(t) ∈ ℝd the value of ! at t

!t = !∣ [0,t] = (!(u), 0 ≤ u ≤ t) ∈ D([0, t],ℝd) therestriction of ! to [0, t].

We will also denote !t− the function on [0, t] given by

!t−(u) = !(u) u < t !t−(t) = !(t−)

For a process X we shall similarly denote

X (t) its value and

Xt = (X (u), 0 ≤ u ≤ t) its path on [0, t].






Properties of functionals of such processes-integration by partsformulae, sensitivity analysis, existence of densities- have beenstudied using the Malliavin calculus.Its starting point is a path space E , and a Wiener/Poisson measureon E ; but the notion of non-anticipativity with respect to afiltration ℱt does not play a role in the construction.We extend the Ito calculus in a non-anticipative manner to alarge class of adapted path-dependent functionals defined on thespace cadlag paths.Our construction builds on Follmer’s (1979) pathwise approach toIto calculus and a pathwise functional derivative proposed by BDupire (2009).The construction is not based on Gaussian properties of theWiener process nor on regularity of semigroups and is carried outfor a general cadlag semimartingale.






Outline

We define a pathwise derivative ∇!F for non-anticipative familiesF = (Ft)t∈[0,T ] of functionals Ft : (D([0, t],ℝd), ∥.∥∞)→ ℝ.

Using this pathwise derivative, we derive a functional change ofvariable formula for processes of the type

Y (t) = Ft({X (u), 0 ≤ u ≤ t}) = Ft(Xt)

where X is a cadlag semimartingale.

This pathwise derivative admits a closure ∇X on the space ofsquare integrable stochastic integrals w.r.t. X , which is shown toverify an integration by parts formula.

We derive a martingale representation formula and an integration byparts formula for stochastic integrals, in terms of ∇X .






Outline

I: Pathwise calculus for non-anticipative functionals.

II: An Ito formula for functionals of semimartingales.

III: Martingale representation formula. Weak derivative.

IV: Relation with Malliavin calculus.

V: Functional Kolmogorov equations.






Functional representation of non-anticipative processes

A process Y adapted to ℱt may be represented as a family offunctionals

Y (t, .) : Ω = D([0, t],ℝd) 7→ ℝ

with the property that Y (t, .) only depends on the path stopped att: Y (t, !) = Y (t, !(. ∧ t) ) so

!∣[0,t] = !′∣[0,t] ⇒ Y (t, !) = Y (t, !′)

Denoting !t = !∣[0,t], we can thus represent Y as

Y (t, !) = Ft(!t)for some Ft : D([0, t],ℝd)→ ℝ

which is ℱt-measurable.Rama Cont & David Fournie Functional Ito calculus





Non-anticipative functionals of cadlag paths

This motivates the following definition:

Definition (Non-anticipative functional)

A non-anticipative functional on the (canonical) path spaceD([0,T ],ℝd) is a family F = (Ft)t∈[0,T ] where

Ft : D([0, t],ℝd)→ ℝ

is ℱt-measurable.

F = (Ft)t∈[0,T ] may be viewed as a functional on the vector bundle∪t∈[0,T ] D([0, t],ℝd).






Functional representation of predictable processes

An ℱt-predictable process Y may be represented as a family offunctionals

Y (t, .) : Ω = D([0,T ],ℝd) 7→ ℝ

with the property

!∣[0,t[ = !′∣[0,t[ ⇒ Y (t, !) = Y (t, !′)

We can thus represent Y as

Y (t, !) = Ft(!t−) for some Ft : D([0, t],ℝd)→ ℝ

where !t−(u) = !(u), u < t and !t−(t) = !(t−).So: an ℱt-predictable Y can be represented as Y (t, !) = Ft(!t−)for some non-anticipative functional F .Ex: integral functionals Y (t, !) =

∫ t0 g(!(u))�(u)du






Functional representation of non-anticipative processes

Any process Y adapted to ℱt = ℱXt may be represented as

Y (t) = Ft({X (u), 0 ≤ u ≤ t}) = Ft(Xt)

where the functional Ft : D([0, t],ℝd)→ ℝ represents thedependence of Y (t) on the path of X on [0, t].F = (Ft)t∈[0,T ] may then be viewed as a functional on the vector

bundle Υ =∪

t∈[0,T ] D([0, t],ℝd).






Horizontal extension of a path

0 0.2 0.4 0.6 0.8 1 1.2−0.5

0

0.5

1

1.5

2

2.5

3

t






Horizontal extension of a path

Let ! ∈ D([0,T ]× Rd), !t ∈ D([0,T ]× Rd) its restriction to[0, t].For h ≥ 0, the horizontal extension !t,h ∈ D([0, t + h],ℝd) of !t

to [0, t + h] is defined as

!t,h(u) = !(u) u ∈ [0, t] ; !t,h(u) = !(t) u ∈]t, t + h]






d∞ metric on Υ =∪

t∈[0,T ] D([0, t],ℝd )

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

!′ ∈ D([0, t + h],ℝd)× S+t+h

d∞( !, !′ ) = supu∈[0,t+h]

∣!t,h(u)− !′(u)∣+ h

On D([0, t],ℝd), d∞ coincides with the supremum norm.






Continuity for non-anticipative functionals

A non-anticipative functional F = (Ft)t∈[0,T ] is said to becontinuous at fixed times if for all t ∈ [0,T [,Ft : D([0, t],ℝd)→ ℝ is continuous w.r.t. the supremum norm.

Definition (Left-continuous functionals)

Define ℂ0,0l as the set of non-anticipative functionals

F = (Ft , t ∈ [0,T [) continuous w.r.t. ! at fixed times andleft-continuous in t:

∀t ∈ [0,T [, ∀� > 0,∀! ∈ D([0, t],ℝd),

∃� > 0, ∀h ∈ [0, t], ∀!′ ∈ ∀! ∈ D([0, t − h],ℝd),

d∞(!, !′) < � ⇒ ∣Ft(!t)− Ft−h(!′)∣ < �






Boundedness-preserving functionals

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

Definition (Boundedness-preserving functionals)

Define B([0,T )) as the set of non-anticipative functionals F onΥ([0,T ]) such that for every compact subset K of ℝd , and t0 < T

∃CK ,t0 > 0, ∀t ≤ t0, ∀! ∈ D([0, t],K ), ∣Ft(!t)∣ ≤ CK ,t0






Measurability and continuity

A non-anticipative functional F = (Ft) applied to X generates anℱt−adapted process

Y (t) = Ft({X (u), 0 ≤ u ≤ t}) = Ft(Xt)

Theorem

Let ! ∈ D([0,T ],ℝd). ,

If F ∈ ℂ0,0l , the path t 7→ Ft(!t−) is left-continuous and

Y (t) = Ft(Xt) defines an optional process.

If F ∈ ℂ0,0l , Y (t) = Ft(Xt) is a predictable process.






Horizontal derivativeVertical derivative of a functionalSpaces of regular functionalsExamplesObstructions to regularityNon-uniqueness of functional representation

Horizontal derivative

Definition (Horizontal derivative)

We will say that the functional F = (Ft)t∈[0,T ] on Υ([0,T ]) is

horizontally differentiable at ! ∈ D([0, t],ℝd) if

DtF (!) = limh→0+

Ft+h(!t,h)− Ft(!t)

hexists

We call DtF (!) the horizontal derivative of F at !.

DF = (DtF )t∈[0,T ] defines a non-anticipative functional.

If Ft(!t) = f (t, !(t)) with f ∈ C 1,1([0,T ]× ℝd) thenDtF (!) = ∂t f (t, !(t)).







Vertical perturbation of a path

0 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: For e ∈ ℝd , the vertical perturbation !et of !t is the cadlag path

obtained by shifting the endpoint:!et (u) = !(u) for u < t and !e

t (t) = !(t) + e.







Definition (Dupire 2009)

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

vertically differentiable at ! ∈ D([0, t]),ℝd) if

ℝd 7→ ℝe → Ft(!

et ) = Ft(!t + e1{t})

is differentiable at 0. Its gradient at 0 is called the verticalderivative of Ft at !

∇!Ft (!) = (∂iFt(!t), i = 1..d) where

∂iFt(!t) = limh→0

Ft(!heit )− Ft(!t)

h







Vertical derivative of a non-anticipative functional

∇!Ft (!).e is simply a directional derivative of Ft in thedirection of the indicator function 1{t}e.

If Ft : D([0, t],ℝd) 7→ ℝ has Frechet derivative DxFt then ithas vertical derivative ∇!Ft(!t) =< DxFt , 1{t} >. However,F may be vertically differentiable without Ft being Frechetdifferentiable for any t ∈ [0,T ].

Note that to compute ∇!Ft (!) we need to compute Foutside C0: even if ! ∈ C0, !h

t /∈ C0.

∇!Ft (!) is ’local’ in the sense that it is computed for t fixedand involves perturbating the endpoint of paths ending at t.







Spaces of differentiable functionals

Definition (Spaces of differentiable functionals)

For j , k ≥ 1 define ℂj ,kb ([0,T ]) as the set of functionals F ∈ ℂ0,0

lwhich are differentiable j times horizontally and k times verticallyat all ! ∈ D([0, t],ℝd), t < T , with

horizontal derivatives Dmt F ,m ≤ j continuous on

(D([0, t],ℝd), ∥.∥∞) for each t ∈ [0,T [

left-continuous vertical derivatives: ∀n ≤ k ,∇n!F ∈ ℂ0,0

l .

Dmt F ,∇n

!F ∈ B([0,T ]).

We can have F ∈ ℂ1,1b ([0,T ]) while Ft not Frechet differentiable

for any t ∈ [0,T ].







Examples of regular functionals

Example (Cylindrical functionals)

For g ∈ C0(ℝd),

Ft(!t) = [!(t)− !(tn−)] 1t≥tn g(!(t1−), !(t2−)..., !(tn−))

is in ℂ1,2b

More generally:

Ft(!t) = u(!(t)−!(tn−)) 1t≥tn g(!(t1−), !(t2−)..., !(tn−))

where u ∈ C 2(ℝd ,ℝ) with u(0) = 0.







Obstructions to regularity

Example (Jump of ! at the current time)

Ft(!t) = !(t)− !(t−) has regular pathwise derivatives:

DtF (!t) = 0 ∇!Ft(!t) = 1

But F /∈ ℂ0,0l .

Example (Jump of ! at a fixed time)

Ft(!t) = 1t≥t0(!(t0)− !(t0−))F ∈ ℂ0,0 has horizontal and vertical derivatives at any order, but∇!Ft(!t) = 1t=t0 fails to be left (or right) continuous.







Obstructions to regularity

Example (Maximum)

Ft(!t) = sups≤t !(s)F ∈ ℂ0,0 but is not vertically differentiable on

{! ∈ D([0, t],ℝd), !(t) = sups≤t

!(s)}.







Derivatives of functionals defined on continuous paths

Theorem

If F 1,F 2 ∈ ℂ1,1 coincide on continuous paths

∀t < T , ∀! ∈ C0([0, t],ℝd),

F 1t (!t) = F 2

t (!t)

then their pathwise derivatives also coincide:

∀t < T , ∀! ∈ C0([0, t],ℝd),

∇!F 1t (!t) = ∇!F 2

t (!t), DtF1t (!t) = DtF

2t (!t)






Quadratic variation for cadlag paths

Follmer (1979): a path f ∈ D([0,T ],ℝ) is said to have finitequadratic variation along a subdivision

�n = (0 = tn0 < ..tk(n)n = T ) if the measures:

�n =

k(n)−1∑i=0

(f (tni+1)− f (tni ))2�tni

where �t is the Dirac measure at t, converge weakly to a Radonmeasure � on [0,T ] such that

[f ](t) = �([0, t]) = [f ]c(t) +∑

0<s≤t(Δf (s))2

where [f ]c is the continuous part of [f ]. [f ] is called the quadraticvariation of f along the sequence (�n).






Change of variable formula for cadlag paths

Let ! ∈ D([0,T ]× ℝd) where ! has finite quadratic variationalong (�n) and

supt∈[0,T ]−�n

∣!(t)− !(t−)∣ → 0

Denote

!n(t) =

k(n)−1∑i=0

!(ti+1−)1[ti ,ti+1)(t) + !(T )1{T}(t)

hni = tni+1 − tni






A pathwise change of variable formula for functionals

Theorem

For any F ∈ ℂ1,2b ([0,T [), the Follmer integral, defined as

∫ T

0

∇!Ft(!t−)d�! := limn→∞

k(n)−1∑i=0

∇!Ftni(!

n,Δ!(tni )tni −

)(!(tni+1)− !(tni ))

exists and FT (!T )− F0(!0) =

∫ T

0

DtFt(!u−)du

+

∫ T

0

1

2tr(∇2!Ft(!u−)d [!]c(u)

)+

∫ T

0

∇!Ft(!t−)d�!

+∑

u∈]0,T ]

[Fu(!u)− Fu(!u−)−∇!Fu(!u−).Δ!(u)]






This pathwise formula implies a functional Ito formula forsemimartingales:

Theorem (Functional change of variable formula)

Let F ∈ ℂ1,2b ([0,T [) be a “smooth” non-anticipative functional.

Ft(Xt)− F0(X0) =

∫ t

0DuF (Xu)du +∫ t

0∇!Fu(Xu).dX (u) +

∫ t

0

1

2tr(t∇2

!Fu(Xu) d [X ](u))

+∑

u∈]0,T ]

[Fu(Xu)− Fu(Xu−) − ∇!Fu(Xu−).ΔX (u)] a.s.

In particular, Y (t) = Ft(Xt) is a semimartingale.






Functional Ito formula

If Ft(Xt) = f (t,X (t)) where f ∈ C 1,2([0,T ]× ℝd) thisreduces to the standard Ito formula.

Y = F (X ) depends on F and its derivatives only via theirvalues on continuous paths: Y can be reconstructed from thesecond-order jet of F on Υc =

∪t∈[0,T ] C0([0, t],ℝd) ⊂ Υ.






Sketch of proof

Consider first a cadlag piecewise constant process:

X (t) =n∑

k=1

1[tk ,tk+1[(t)�k �k ℱtk −measurable

Each path of X is a sequence of horizontal, vertical moves andjumps:

Xtk+1= (Xtk ,hk )�k+1−�k hk = tk+1 − tk

Ftk+1(Xtk+1

)− Ftk (Xtk ) =

Ftk+1(Xtk+1

)− Ftk+1(Xtk+1−) jump

+Ftk+1(Xtk+1

)− Ftk+1(Xtk ,hk ) vertical move

+Ftk+1(Xtk ,hk )− Ftk (Xtk ) horizontal move






Sketch of proof

Horizontal step: fundamental theorem of calculus for�(h) = Ftk+h(Xtk ,h)

Ftk+1(Xtk ,hk )− Ftk (Xtk )

= �(hk)− �(0) =

∫ tk+1

tk

DtF (Xt)dt

Vertical step: apply Ito formula to (u) = Ftk+1(X u

tk ,hk)

Ftk+1(Xtk+1

)− Ftk+1(Xtk ,hk ) = (X (tk+1)− X (tk))− (0)

=

∫ tk+1

tk

∇!Ft(Xt).dX +1

2tr(∇2

!Ft(Xt)d [X ])






Sketch of proof

General case: approximate X by a sequence of simple predictableprocesses nX with nX (0) = X (0). For each n ≥ 1,

FT (nXT )− F0(X0) =

∫ T

0DtF (nXt)dt +

∫ T

0∇!F (nXt).dX

∑ΔnX (t)∕=0

F (Xt)− F (Xt−)−ΔX (t).∇!F (Xt−) +1

2

∫ T

0tr∇2

!F (nXt)d [X ]

The ℂ1,2b assumption on F implies that all derivatives involved in

the expression are left continuous in d∞ metric, which allows tocontrol their convergence as n→∞ using dominated convergence+ the dominated convergence theorem for stochastic integrals.






Extension: locally regular functional

If (�n)n≥0 is an increasing sequence of optional times with �n → Ta.s. and F n ∈ ℂ1,2

b ([0,T [) such that

Ft(!) =∑n≥1

1[�n(!),�n+1(!)[(t) F nt (!)

then Y (t) = Ft(Xt) is a semimartingale and

Ft(Xt)− F0(X0) =

∫ t

0DuF (Xu)du +∫ t

0∇!Fu(Xu).dX (u) +

∫ t

0

1

2tr(t∇2

!Fu(Xu) d [X ](u))

+∑

u∈]0,T ]

[Fu(Xu)− Fu(Xu−) − ∇!Fu(Xu−).ΔX (u)] a.s.

This result covers in particular functionals involving exit times.Rama Cont & David Fournie Functional Ito calculus





Tangent process: intrinsic definition

Definition (Vertical derivative of a process)

Define C1,2b (X ) the set of processes Y which admit a functional

representation in ℂ1,2b in terms of X :

C1,2b (X ) = {Y , ∃F ∈ ℂ1,2

b ([0,T ]), Y (t) = Ft(Xt) a.s.}

If det(�.�∗) > 0 a.s. then for Y ∈ C1,2b (X ), the process:

∇XY (t) = ∇!Ft(Xt)

is uniquely defined up to an evanescent set, independently of thechoice of F ∈ ℂ1,2

b . We call ∇XY the vertical derivative of Ywith respect to X .






Vertical derivative for Brownian functionals

In particular when X is a standard Brownian motion, A = Id :

Definition

Let W be a standard d-dimensional Brownian motion. For anyY ∈ C1,2

b (W ) with representation Y (t) = Ft(Wt), the predictableprocess

∇W Y (t) = ∇!Ft(Wt)

is uniquely defined up to an evanescent set, independently of thechoice of the representation F ∈ ℂ1,2

b .






Martingale representation theorem

Consider now the case whereX (t) =

∫ t0 �(t).dW (t) +

∫ t0

∫zJX (dtdz) is a martingale. 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 ]. It iswell known (Ito, Jacod-Yor) that

Y (T ) = E [Y (T )] +∫ T

0 �dX +∫ T

0

∫ℝd (t, z)JX (dtdz)

for some predictable process � and predictable random function measurable with respect to the ℙ-completed filtration �(N ∨ ℱt).Many applications –stochastic control, solution of BSDEs, hedgingformulas in mathematical finance– involve the computation of �, .






Martingale representation formula

Let Y (t) = E [H∣ℱt ] where H = H(X (t), t ∈ [0,T ]) withE [∣H∣2] <∞

Theorem

If Y ∈ C1,2b (X ) then

Y (T ) = E [Y (T )] +∫ T

0

�(t)︷︸︸︷∇XY (t) dX (t)

+∫ T

0

∫ℝd (Ft(Xt− + z1{t})− Ft(Xt−))︸︷︷︸

(t,z)

JX (dtdz)

In particular �, have regular, ℱt−adapted versions (no need tocomplete).






Pathwise computation of martingale representations

Consider for example the case where X is solution of a stochasticdifferential equation. Then we can numerically simulate X .Let nX be the solution of a n-step Euler scheme and Yn a MonteCarlo estimator of Y obtained using nX .

Compute the Monte Carlo estimator Yn(t, nX ht (!))

Bump the endpoint by h.

Recompute the Monte Carlo estimator Yn(t, nX ht (!)) (with

the same simulated paths)

An estimator of the integrand is given by

�n(t, !) :=Yn(t, nX h

t (!))− Yn(t, nX ht (!))

h






Pathwise computation of martingale representations

�n(t, !) ≃ Yn(t, nX ht (!))− Yn(t, nX h

t (!))

h

For a general C1,2b (S) + some mild regularity assumptions

∀1/2 > � > 0, n1/2−�∣�n(t)− �(t)∣ → 0 ℙ− a.s.

This rate is attained for a time-step size h = cn−1/4+�/2.






An integration by parts formulaMartingale Sobolev spaceWeak derivativeRelation with Malliavin derivative

ℐ2(X ) = {∫ .

0 �dX , � ℱt − adapted, E [∫ T

0 ∥�(t)∥2d [X ](t)] <∞} = {

∫ .0 �dX , � ∈ ℒ2(X )}

Theorem (Non-anticipative integration by parts formula)

Let Y ∈ C1,2b (X ) be a (ℙ, (ℱt))-martingale with Y (0) = 0 and �

an ℱt−adapted process with E [∫ T

0 ∥�(t)∥2d [X ](t)] <∞. Then

E

(Y (T )

∫ T

0�dX

)= E

(∫ T

0∇XY .�d [X ]

)This allows to extend the functional Ito formula to the closure ofC1,2b (X ) ∩ ℐ2(X ) wrt to the norm

E∥Y (T )∥2 = E [

∫ T

0∥∇XY (t)∥2d [X ](t) ]







A “Martingale Sobolev space”

Definition (Martingale Sobolev space)

Define W1,2(X ) as the closure in ℐ2(X ) ofD(X ) = C1,2

b (X ) ∩ ℐ2(X ).

Lemma

{∇XY ,Y ∈ D(X )} is dense in ℒ2(X ) and

W1,2(X ) = {∫ .

0�dX , E

∫ T

0∥�∥2d [X ] <∞}.

So W1,2(X )=all square-integrable integrals with respect to X .







Weak derivative for stochastic integrals

Theorem (Weak derivative on W1,2(X ))

The vertical derivative ∇X : D(X ) 7→ ℒ2(X ) is closable onW1,2(X ). Its closure defines a bijective isometry

∇X : W1,2(X ) 7→ ℒ2(X )∫ .

0�.dX 7→ �

characterized by the following integration by parts formula: forY ∈ W1,2(X ), ∇XY is the unique element of ℒ2(X ) such that

∀Z ∈ D(X ), E [Y (T )Z (T )] = E

[∫ T

0∇XY (t)∇XZ (t)d [X ](t)

].

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

IX : ℒ2(X ) 7→ W1,2(X )

� 7→∫ .

0�.dX

in the following sense:

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

i .e. E [Y (T )

∫ T

0�.dX ] = E [

∫ T

0∇XY �.d [X ] ]







Vertical derivative as inverse of Ito’s integral

We have thus shown that the (weak) vertical derivative ∇X acts asthe inverse of the Ito stochastic integral on W1,2(X ):

If Y (t) =

∫ t

0�.dX then ∇XY = �

i.e. ∇X acts as a “stochastic derivative” (Zabczyk 1978).This extends some constructions of “stochastic derivatives” forBrownian motion and continuous martingales (Isaacson 1973,Zabczyk 1978, Davis 1979, Yoeurp 1981).







Computation of the weak derivative

For D(X ) = C1,2b (X ) ∩ ℐ2(X ), the weak derivative may be

computed pathwise:

∀Y ∈ C1,2b (X ), ∇XY (t, !) = lim

h→0

Y (t,Xt(!))− Y (t,Xt(!))

h

For Y ∈ W1,2(X ), ∇XY = limn∇XY n where Y n ∈ D(X ) with

E∥Y n(T )− Y (T )∥2 n→∞→ 0

If Y n is an estimator of Y computed from a discretization schemefor X then one may use the following estimator for � = ∇XY :

∇XY (t,Xt(!)) =Y n(t,X

h(n)t (!))− Y n(t,Xt(!))

h(n)

where h(n) ∼ cn−� is chosen according to the “smoothness” ofthe functional Y .







Relation with Malliavin derivative

Consider the case where X = W . Then for Y ∈ W1,2(W )

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

∫ T

0∇W Y (t)dW (t)

If H = Y (T ) is Malliavin-differentiable e.g. H = Y (T ) ∈ D1,1

then the Clark-Haussmann-Ocone formula implies

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

∫ T

0

pE [DtH∣ℱt ]dW (t)

where D is the Malliavin derivative.








Theorem (Intertwining formula)

Let Y be a (ℙ, (ℱWt )t∈[0,T ]) martingale. If Y ∈ C1,2(W ) and

Y (T ) = H ∈ D1,2 then

E [DtH∣ℱt ] = (∇W Y )(t) dt × dℙ− a.e.

i.e. the conditional expectation operator intertwines ∇W and D:

E [DtH∣ℱt ] = ∇W (E [H∣ℱt ]) dt × dℙ− a.e.








The following diagram is commutative, in the sense of dt × dℙalmost everywhere equality:

W1,2(W )∇W→ ℒ2(W )

↑(E [.∣ℱt ])t∈[0,T ] ↑(E [.∣ℱt ])t∈[0,T ]

D1,2 D→ L2([0,T ]× Ω)

Note however that ∇X may be constructed for any Ito-Levymartingale X and its construction does not involve Gaussianproperties of X .






Functional characterization of martingales

Functional equation for martingales

Consider now a ℝd -valued Levy process L with characteristic triplet(b,A, �)Denote supp(L) the topological support of the law of L in(D([0,T ],ℝd), ∥.∥∞):

supp(L) = {!, any neighborhood V of !, ℙ(L ∈ V ) > 0}

The topological support may be characterized in terms of (b,A, �)see e.g. Th Simon (2002)







Functional PIDE for martingales

Theorem (Harmonic functionals of a Levy process)

Let F ∈ ℂ1,2b . Then Y (t) = Ft(Lt) is a local martingale if and

only if F satisfies the following functional equation:∫ℝd

(Ft(!t− + z1{t})− Ft(!t−)− 1∣z∣≤1 z .∇!Ft(Xt−)

)�(dz) +

DtF (!t) + b.∇!Ft(!t) +1

2tr[∇2

!F (!t).A] = 0

for ! ∈supp(X ).

We call such functionals X−harmonic functionals.







Structure equation for Brownian martingales

In particular when X = W is a d-dimensional Wiener process, weobtain a characterization of ‘regular’ Brownian local martingales assolutions of a functional PDE:

Theorem

Let F ∈ ℂ1,2b . Then Y (t) = Ft(Wt) is a local martingale on [0,T ]

if and only if

∀t ∈ [0,T ], ! ∈ C0([0,T ],ℝd),

DtF (!t) +1

2tr(∇2!F (!t)

)= 0.







Theorem (Uniqueness of solutions)

Let h be a continuous functional on (D([0,T ]), ∥.∥∞). Then anyF ∈ ℂ1,2

b verifying ∀! ∈ D([0,T ]),∫ℝd

(Ft(!t− + z1{t})− Ft(!t−)− 1∣z∣≤1 z .∇!Ft(Xt−)

)�(dz) +

DtF (!t) + b∇!Ft(!t) +1

2tr[∇2

!F (!t).A] = 0

FT (!) = h(!), E [ supt∈[0,T ]

∣Ft(Xt)∣] <∞

is uniquely defined on the topological support of X : ifF 1,F 2 ∈ ℂ1,2

b ([0,T ]) are two solutions then

∀! ∈ supp(X ), ∀t ∈ [0,T ], F 1t (!t) = F 2

t (!t).







Extensions and applications

The results can be extended to functionals of cadlagsemimartingales and Dirichlet processes with cadlag paths andfunctionals of the quadratic variation process (JFA 2010)

All results (except global uniqueness for Functional PDE) canbe localized using stopping times: important for applying tofunctionals involving stopped processes/ exit times.

Application: pathwise maximum principle for non-Markoviancontrol problems.

Infinite-dimensional extension to functionals of aBanach-valued process.

Application: hedging of path-dependent derivatives.







References

R. Cont & D Fournie (2010) A functional extension of the Itoformula, Comptes Rendus Acad. Sci. Paris, Ser. I , Vol. 348,51-67, Jan 2010.

R. Cont & D Fournie (2010) Functional Ito formula andstochastic integral representation of martingales,arxiv.org/math.PR.

R. Cont & D Fournie (2010) Change of variable formulas fornon-anticipative functionals on path space, Journal ofFunctional Analysis, Vol 259, 1043–1072.


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