An Introduction to Malliavin calculus and its applications Lecture 1 ...

An Introduction to Malliavin calculus and itsapplications

Lecture 1: Derivative and divergence operators

David Nualart

Department of MathematicsKansas University

University of WyomingSummer School 2014

David Nualart (Kansas University) May 2014 1 / 24

Malliavin Calculus

Paul Malliavin (1925-2010) introduced in the 70’s a calculus ofvariations with respect to the trajectories of Brownian motionThe purpose of this calculus is to provide a probabilistic proof ofHormander’s hypoellipticity theorem


Finite dimensional case

The Malliavin calculus is a differential calculus in a Gaussianprobability space.

Consider first the finite dimensional case. The probability space(Ω,F ,P) is

Ω = Rn

F = B(Rn)

P is the standard Gaussian probability with density

p(x) =1

(2π)n/2 e−|x |2/2.


Two differential operators

(i) Derivative operator :

∇F =

(∂F∂x1

, . . . ,∂F∂xn

),

where F : Rn → R.(ii) Divergence operator :

δ(u) =n∑

i=1

(uixi −

∂ui

∂xi

)= 〈u, x〉 − divu,

where u : Rn → Rn.


Propositionδ is the adjoint of ∇, that is,

E(〈u,∇F 〉) = E(Fδ(u)),

if F and u are continuously differentiable and their partial derivatives have atmost polynomial growth.

Proof :Integrating by parts, and using ∂p

∂xi= −xip(x) we obtain

∫Rn〈∇F ,u〉p(x)dx =

n∑i=1

∫Rn

∂F∂xi

uip(x)dx

=n∑

i=1

(−∫Rn

F∂ui

∂xip(x)dx +

∫Rn

Fuixip(x)dx)

=

∫Rn

Fδ(u)p(x)dx .


The Wiener space

Consider the probability space (Ω,F ,P), where

Ω = C([0,T ])

F is the Borel σ-field of Ω

P is the Wiener measure : For all 0 ≤ t1 < · · · < tn, ai < bi ,

P ω : ai ≤ ω(ti) ≤ bi ,1 ≤ i ≤ n

=

∫ bn

an

· · ·∫ b1

a1

n∏i=1

1√2π(ti − ti−1)

e−

(xi−xi−1)2

2(ti−ti−1) dx1 · · · dxn,

with the convention t0 = 0 and x0 = 0.


The canonical stochastic process Wt (ω) = ω(t), t ∈ [0,T ] is aBrownian motion :

W0 = 0.For all 0 ≤ s < t , Wt −Ws has the normal law N(0, t − s), withdensity 1√

2π(t−s)e−x2/(2(t−s)).

For all 0 ≤ t0 < t1 < · · · < tn, the incrementsWt1 −Wt0 , . . . ,Wtn −Wtn−1 are independent random variables.


Derivative operatorLet F : Ω→ R. The derivative DF takes values in H = L2([0,T ]). That is,DtF ,T ∈ [0,T ] is a stochastic process.

For any h ∈ H, we denote by W (h) the Wiener integralW (h) =

∫ T0 h(t)dW (t).

Let S be the set of smooth and cylindrical random variables of the form

F = f (W (h1) . . . ,W (hn)),

where f ∈ C∞p (Rn) (f and all ist partial derivatives have polynomialgrowth) and hi ∈ H.

DefinitionFor F ∈ S the derivative DF is the H-valued random variable defined by

DtF =n∑

i=1

∂f∂xi

(W (h1) . . . ,W (hn))hi (t).

Examples : D(W (h)) = h, D(Wt1 ) = 1[0,t1].David Nualart (Kansas University) May 2014 8 / 24

The Cameron-Martin space H1 ⊂ Ω is the set of functions of theform ψ(t) =

∫ t0 h(s)ds, where h ∈ H.

For ant h ∈ H, 〈DF ,h〉H is the derivative of F in the direction of∫ ·0 h(s)ds :

〈DF ,h〉H =

∫ T

0htDtFdt =

ddε

F(ω + ε

∫ ·0

hsds)|ε=0.

Example : If F = Wt1

F(ω + ε

∫ ·0

hsds)

= ω(t1) + ε

∫ t1

0hsds,

so, 〈DF ,h〉H =∫ t1

0 hsds, and DtF = 1[0,t1](t).


Divergence operator

Let u ∈ SH be an smooth and cylindrical stochastic process of the form

u(t) =n∑

j=1

Fjhj (t),

where the Fj ∈ S, and hj ∈ H.

DefinitionWe define the divergence of u, δ(u) as the random variable

δ(u) =n∑

j=1

FjW (hj )−n∑

j=1

⟨DFj ,hj

⟩H .

Example : δ(h) = W (h).


PropositionLet F ∈ S and u ∈ SH . Then,

E(Fδ(u)) = E(〈DF ,u〉H).

Proof : We can assume that

F = f (W (h1) . . . ,W (hn))

and

u =n∑

j=1

g(W (h1) . . . ,W (hn))hj ,

where h1, . . . ,hn are orthonormal elements in H. In this case, theduality relationship reduces to the finite dimensional case.


Notation : DhF = 〈DF ,h〉H , for any h ∈ H and F ∈ S.

PropositionSuppose that u, v ∈ SH , F ∈ S and h ∈ H. Then, if ei , i ≥ 1 is acomplete orthonormal system in H we have

E(δ(u)δ(v)) = E(〈u, v〉H) + E

∞∑i,j=1

Dei 〈u,ej〉HDej 〈v ,ei〉H

, (1)

Dh(δ(u)) = δ(Dhu) + 〈h,u〉H , (2)δ(Fu) = Fδ(u)− 〈DF ,u〉H . (3)

Property (1) can also be written as

E [δ(u)δ(v)] = E

[∫ T

0utvtdt

]+ E

[∫ T

0

∫ T

0DsutDtvsdsdt

].


Proof of Dh(δ(u)) = δ(Dhu) + 〈h,u〉H :

Assume u =∑n

j=1 Fjhj . Then, using Dh(W (hj) = 〈h,hj〉H , we obtain

Dh(δ(u)) = Dh

n∑j=1

FjW (hj)−n∑

j=1

〈DFj ,hj〉H

=

n∑j=1

Fj⟨h,hj

⟩H +

n∑j=1

(DhFjW (hj)−

⟨Dh(DFj

),hj⟩

H

)= 〈u,h〉H + δ(Dhu).


Proof of E(δ(u)δ(v)) = E(〈u, v〉H) + E

∞∑i,j=1

Dei 〈u,ej〉HDej 〈v ,ei〉H

:

Using the duality formula and property (2) yields

E (δ(u)δ(v))) = E (〈v ,D(δ(u))〉H)

= E

( ∞∑i=1

〈v ,ei〉H Dei (δ(u))

)

= E

( ∞∑i=1

〈v ,ei〉H (〈u,ei〉H + δ(Dei u))

)

= E (〈u, v〉H) + E

∞∑i,j=1

Dei

⟨u,ej

⟩H Dej 〈v ,ei〉H

.


Proof of δ(Fu) = Fδ(u)− 〈DF ,u〉H :

For any smooth random variable G ∈ S we have,

D(FG) = FDG + GDF .

Then, using the duality relationship

E [δ(u)G] = E (〈DG,Fu〉H)

= E (〈u,D(FG)−GDF 〉H)

= E ((δ(u)F − 〈u,DF 〉H) G) .

This implies the result because S is dense in L2(Ω).


Sobolev spaces

PropositionThe operator D is closable from Lp(Ω) to Lp(Ω; H) for any p ≥ 1.

Proof : Assume FN ∈ S satisfies

FNLp(Ω)−→ 0, DFN

Lp(Ω;H)−→ η.

Then, η = 0. Indeed, for any u =∑N

j=1 Gjhj ∈ SH such that Gj and DGjare bounded, we have

E(〈η,u〉H) = limN→∞

E(〈DFN ,u〉H)

= limN→∞

E(FNδ(u)) = 0.

This implies that η = 0.


For p ≥ 1, we denote by D1,p the closure of S with respect to theseminorm

‖F‖1,p =

E [|F |p] + E

∣∣∣∣∣∫ T

0(DtF )2dt

∣∣∣∣∣p/21/p

.

For p = 2, the space D1,2 is a Hilbert space with the scalar product

〈F ,G〉1,2 = E(FG) + E

[∫ T

0DtFDtGdt

].

In the same way we can introduce the spaces D1,p(H) by takingthe closure of SH .


DefinitionThe domain of the divergence operator Domδ in L2(Ω) is the set ofprocesses u ∈ L2(Ω× [0,T ]) such that there exists δ(u) ∈ L2(Ω)satisfying the duality relationship

E(〈DF ,u〉H) = E(δ(u)F ),

for any F ∈ D1,2.

Clearly if un ∈ SH satisfies unL2(Ω;H)−→ u and δ(un)

L2(Ω)−→ G, then ubelongs to Domδ and δ(u) = G.


Property (1) holds for u, v ∈ D1,2(H) ⊂ Domδ and

E(δ(u)2) ≤ E

[∫ T

0(ut )

2dt

]+ E

[∫ T

0

∫ T

0(Dsut )

2dsdt

]= ‖u‖2

1,2,H .

Property (2) :Dh(δ(u)) = δ(Dhu) + 〈h,u〉H

holds if u ∈ D1,2(H) and Dhu is in the domain of δ.

Property (3) :δ(Fu) = Fδ(u)− 〈DF ,u〉H

holds if F ∈ D1,2, Fu ∈ L2(Ω; H), u ∈ Domδ and the right-hand side existsand is square integrable.


Meyer inequalities

TheoremFor any p > 1 and u ∈ D1,p(H),

E(|δ(u)|p) ≤ cp

(E(‖Du‖pH⊗H) + E(‖u‖pH)

).

1 Proved first by Paul-Andre Meyer (1934-2003)2 A more modern proof is based on the boundedness in Lp of the

Riesz transform (Gilles Pisier)


Iterated derivative

The k th derivative Dk F of a random variable F ∈ S is a k -parameterprocess obtained by iteration :

Dkt1,...tk F =

n∑i1,...,ik =1

∂k f∂xi1 · · · ∂xik

f (W (h1), . . .W (hn))hi1 (t1) · · · hik (tk ).

For any p ≥ 1, the operator Dk is closable from Lp(Ω) into Lp(Ω; H⊗k )and we denote by Dk,p the closure of S with respect to the norm

‖F‖k,p =

E [|F |p] + E

k∑j=1

∣∣∣∣∣∫

[0,T ]k(Dj

t1,...,tj F )2dt1 · · · dtj

∣∣∣∣∣p/21/p

.


Denote by Ft the σ-field generated by Ws,0 ≤ s ≤ t.

A stochastic process u = ut , t ∈ [0,T ] is adapted if for each t ∈ [0,T ],ut is Ft -measurable.

We denote by L2a the subspace of L2(Ω× [0,T ]) formed by adapted

processes.

Theorem (Gaveau-Trauber 1982)L2

a ⊂ Domδ and for any u ∈ L2a, δ(u) coincides with the Ito’s stochastic

integral :

δ(u) =

∫ T

0utdWt

If u not adapted δ(u) coincides with an anticipating stochastic integralintroduced by Skorohod in 1975.

Using techniques of Malliavin calculus, Nualart-Pardoux 1988 developeda stochastic calculus for the Skorohod integral.


Proof :

(i) If u =∑n

j=1 Fj1[aj ,bj ] and Fj is a random variable in S, Faj -measurable,then δ(u) coincides with the Ito integral of u because

δ(u) =n∑

j=1

Fj (W (bj )−W (aj )−n∑

j=1

∫ bj

aj

DtFdt =n∑

j=1

Fj (W (bj )−W (aj ),

and DtF = 0 if t > aj .

(ii) The result follows by approximating any square integrable adaptedprocess by cylindrical adapted smooth processes.


If u and v are adapted, then for s < t , Dtvs = 0 and for s > t ,Dsut = 0 and Property (1) leads to the isometry property

E [δ(u)δ(v)] = E

[∫ T

0utvtdt

].

If u is an adapted process in D1,2(H), then from Property (2) weobtain

Dt

(∫ T

0usdWs

)= ut +

∫ T

tDtusdWs,

because Dtus = 0 if t > s.


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