an introduction to malliavin calculus and its applications lecture ii

An Introduction to Malliavin calculus and itsapplications

Lecture II: Wiener chaos and the Ornstein-Uhlenbeck semigroup

David Nualart

Department of MathematicsKansas University

University of WyomingSummer School 2014

David Nualart (Kansas University) May 2014 1 / 22

Multiple stochastic integrals

W = Wt , t ∈ [0,T ] is a Brownian motion defined in the canonicalprobability space (Ω,F ,P).

L2s([0,T ]n) is the space of symmetric square integrable functions

f : [0,T ]n → R.

For any f ∈ L2s([0,T ]n)

‖f‖2L2([0,T ]n) = n!

∫∆n

f 2(t1, . . . , tn)dt1 · · · dtn,

where∆n = (t1, . . . , tn) ∈ [0,T ]n : 0 < t1 < · · · < tn < T.

If f : [0,T ]n → R we define its symmetrization as

f (t1, . . . , tn) =1n!

∑σ

f (tσ(1), . . . , tσ(n)),

where the sum runs over all permutations σ of 1,2, . . . ,n.


The multiple stochastic integral of f ∈ L2s([0,T ]n) is defined as an

iterated Ito integral :

In(f ) = n!

∫ T

0

∫ tn

0· · ·∫ t2

0f (t1, . . . , tn)dWt1 · · · dWtn .

We have the following property :

E [In(f )Im(g)] =

0 if n 6= mn!〈f ,g〉L2([0,T ]n) if n = m.

If f ∈ L2([0,T ]n) is not necessarily symmetric we define

In(f ) = In(f ).


The nth Hermite polynomial is defined by

hn(x) = (−1)nex2/2 dn

dxn (e−x2/2.

The first Hermite poynomials are h0(x) = 1, h1(x) = x ,h2(x) = x2 − 1, h3(x) = x3 − 3x , . . . .For any g ∈ L2([0,T ]) we have

In(g⊗n) = ‖g‖|nhn

(∫ T0 gtdWt

‖g‖

),

where g⊗n(t1, . . . , tn) = g(t1) · · · g(tn).


Product formula

Let f ∈ L2s([0,T ]n), and g ∈ L2

s([0,T ]m). For any r = 0, . . . ,n ∧m, we definethe contraction of f and g of order r to be the element of L2([0,T ]n+m−2r )defined by

(f ⊗r g) (t1, . . . , tn−r , s1, . . . , sm−r )

=

∫[0,T ]r

f (t1, . . . , tn−r , x1, . . . , xr )g(s1, . . . , sm−r , x1, . . . , xr )dx1 · · · dxr .

We denote by f ⊗r g the symmetrization of f ⊗r g.

Product of two multiple stochastic integrals

In(f )Im(g) =n∧m∑r=0

r !

(nr

)(mr

)In+m−2r (f ⊗r g).


Wiener Chaos expansion

TheoremF ∈ L2(Ω) can be uniquely expanded into a sum of multiple stochasticintegrals :

F = E [F ] +∞∑

n=1

In(fn).

For any n ≥ 1 we denote by Hn the closed subspace of L2(Ω) formed byall multiple stochastic integrals of order n. For n = 0, H0 is the space ofconstants. Then, we have the orthogonal decomposition

L2(Ω) = ⊕∞n=0Hn.

The theorem follows from the fact that if a random variable G ∈ L2(Ω) isorthogonal to ⊕∞n=0Hn, then it is orthogonal to all random variables of the

form(∫ T

0 gtdWt

)k, where g ∈ L2([0,T ]), k ≥ 0. This implies that G is

orthogonal to all the exponentials exp(∫ T

0 gtdWt

), which form a total set

in L2(Ω). So G = 0.David Nualart (Kansas University) May 2014 6 / 22

Derivative operator on the Wiener chaosLet f ∈ L2

s([0,T ]n). Then In(f ) ∈ D1,2, and

Dt In(f ) = nIn−1(f (·, t))

Proof : Assume f = g⊗n, with ‖g‖ = 1. Let θ =∫ T

0 gtdWt . Then

Dt In(f ) = Dt (hn(θ)) = h′n(θ)Dtθ = nhn−1(θ)gt

= ngt In−1(g⊗(n−1)) = nIn−1(f (·, t)).

Moreover

E∫ T

0[Dt In(f )]2dt = n2

∫ T

0E [In−1(f (·, t))2]dt

= n2(n − 1)!

∫ T

0‖f (·, t)‖2

L2([0,T ]n−1)dt

= nn!‖f‖2L2([0,T ]n)

= nE [In(f )2].


PropositionLet F ∈ L2(Ω) with the Wiener chaos expansion F =

∑∞n=0 In(fn). Then

F ∈ D1,2 if and only if

E(‖DF‖2H) =

∞∑n=1

nn!‖fn‖2L2([0,T ]n) <∞,

and in this case

DtF =∞∑

n=1

nIn−1(fn(·, t)).

If F ∈ Dk,2, then

Dkt1,...,tk F =

∞∑n=k

n(n − 1) · · · (n − k + 1)In−k (fn(· , t1, . . . , tk )).

As a consequence, if F ∈ D∞,2 := ∩kDk,2, then (Stroock’s formula)

fn =1n!

E(DnF )


Example : F = W 31 . Then,

f1(t1) = E(Dt1W 31 ) = 3E(W 2

1 )1[0,1](t1) = 31[0,1](t1),

f2(t1, t2) =12

E(D2t1,t2W 3

1 ) = 3E(W1)1[0,1](t1 ∨ t2) = 0,

f3(t1, t2, t3) =16

E(D3t1,t2,t3W 3

1 ) = 1[0,1](t1 ∨ t2 ∨ t3),

and we obtain the Wiener chaos expansion

W 31 = 3W1 + 6

∫ 1

0

∫ t1

0

∫ t2

0dWt1dWt2dWt3 .


Divergence on the Wiener chaos expansion

A square integrable stochastic process u = ut , t ∈ [0,T ], has anorthogonal expansion of the form

ut =∞∑

n=0

In(fn(·, t)),

where f0(t) = E [ut ] and for each n ≥ 1, fn ∈ L2([0,T ]n+1) is asymmetric function in the first n variables.

Proposition

The process u belongs to the domain of δ if and only if the series

δ(u) =∞∑

n=0

In+1(fn) (1)

converges in L2(Ω).


Proof : Suppose that G = In(g) is a multiple stochastic integral of order n ≥ 1,where g is symmetric. Then,

E(〈u,DG〉H) =

∫ T

0E(In−1(fn−1(·, t))nIn−1(g(·, t)))dt

= n(n − 1)!

∫ T

0〈fn−1(·, t),g(·, t)〉L2([0,T ]n−1) dt

= n! 〈fn−1,g〉L2([0,T ]n) = n!⟨

fn−1,g⟩

L2([0,T ]n)

= E(

In(

fn−1

)In(g)

)= E

(In(

fn−1

)G).

If u ∈ Domδ, we deduce that

E(δ(u)G) = E(

In(

fn−1

)G)

for every G ∈ Hn. This implies that In(

fn−1

)coincides with the projection of

δ(u) on the nth Wiener chaos. Consequently, the series in (1) converges inL2(Ω) and its sum is equal to δ(u). The converse can be proved by similararguments.


The Ornstein-Uhlenbeck semigroup

Consider the one-parameter semigroup Tt , t ≥ 0 of contractionoperators on L2(Ω) defined by

Tt (F ) =∞∑

n=0

e−nt In(fn),

where F =∑∞

n=0 In(fn).

Proposition (Mehler’s formula)

Let W ′ = W ′t , t ∈ [0,T ] be an independent copy of W. Then, for any

t ≥ 0 and F ∈ L2(Ω) we have

Tt (F ) = E ′(F (e−tW +√

1− e−2tW ′)), (2)

where E ′ denotes the mathematical expectation with respect to W ′.


Proof : Both Tt and the right-hand side of (2) give rise to linear contractionoperators on L2(Ω). Thus, it suffices to show (2) whenF = exp

(λW (h)− 1

2λ2), where W (h) =

∫ T0 htdWt and h ∈ H is an element of

norm one, and λ ∈ R. We have,

E ′(

exp(

e−tλW (h) +√

1− e−2tλW ′(h)− 12λ2))

= exp(

e−tλW (h)− 12

e−2tλ2)

=∞∑

n=0

e−nt λn

n!hn (W (h)) = TtF ,

because

F =∞∑

n=0

λn

n!hn (W (h))

and

eaz− 12 a2

=∞∑

n=0

an

n!hn(z).


Mehler’s formula implies that the operator Tt is nonnegative.

The oerator Tt is symmetric :

E(GTt (F )) = E(FTt (G)) =∞∑

n=0

e−ntE(In(fn)In(gn)).

Tt , t ≥ 0 is the semigroup of transition probabilities of a Markovprocess with values in C([0,T ]), whose invariant measure in the Wienermeasure. This process can be expressed in terms of a Wiener sheet Was follows :

Xt,τ =√

2∫ t

−∞

∫ τ

0e−(t−s)W (dσ, ds),

τ ∈ [0,T ], t ≥ 0.


Hypercontractivity

If F ∈ Lp(Ω), p > 1 and q(t) = e2t (p − 1) + 1 > p, t > 0, then

‖TtF‖q(t) ≤ ‖F‖p, (3)

Consequences :For any 1 < p < q <∞ the norms ‖ · ‖p and ‖ · ‖q are equivalenton any Wiener chaos Hn.For each n ≥ 1 and 1 < p <∞, the projection on the nth Wienerchaos is bounded in Lp(Ω).


The generator of the Ornstein-Uhlenbeck semigroup

The infinitesimal generator of the semigroup Tt in L2(Ω) is given by

LF = limt↓0

TtF − Ft

=∞∑

n=1

−nIn(fn),

if F =∑∞

n=0 In(fn).The domain of L is

Dom L = F ∈ L2(Ω),∞∑

n=1

n2n!‖fn‖22 <∞ = D2,2.


The next proposition explains the relationship between the operators D, δ,and L.

PropositionLet F ∈ L2(Ω). Then F ∈ Dom L if and only if F ∈ D1,2 and DF ∈ Dom δ, andin this case, we have

δDF = −LF

Proof Let F =∑∞

n=0 In(fn). For any random variable G = Im(gm) we have,using the duality relationship

E(GδDF ) = E(〈DG,DF 〉H) = mm!〈gm, fm〉L2([0,T ]n)

= E

(G∞∑

n=1

nIn(fn)

)= −E(GLF ),

and the result follows easily.


The operator L behaves as a second-order differential operator onsmooth random variables.

Proposition

Suppose that F = (F 1, . . . ,F m) is a random vector whose componentsbelong to D2,4. Let ϕ be a function in C2(Rm) with bounded first andsecond partial derivatives. Then ϕ(F ) ∈ Dom L, and

L(ϕ(F )) =m∑

i,j=1

∂2ϕ

∂xi∂xj(F )〈DF i ,DF j〉H +

m∑i=1

∂ϕ

∂xi(F )LF i .


Proof : Suppose first that F ∈ S is of the form

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

f ∈ C∞p (Rn). Then

DtF =n∑

i=1

∂f∂xi

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

Consequently, DF ∈ SH ⊂ Dom δ and we obtain

δDF =n∑

i=1

∂f∂xi

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

−n∑

i,j=1

∂2f∂xi∂xj

(W (h1), . . . ,W (hn))〈hi ,hj〉H ,

which yields the desired result because L = −δD. In the general case, itsuffices to approximate F by smooth random variables in the norm ‖ · ‖2,4,and ϕ by functions in C∞p (Rm), and use the continuity of the operator L in thenorm ‖ · ‖2,2 .


In the finite-dimensional case (Ω = Rn equipped with the standardGaussian law),

L = ∆− x · ∇

coincides with the generator of the Ornstein-Uhlenbeck processXt , t ≥ 0 in Rn, which is the solution to the stochastic differentialequation

dXt =√

2dWt − Xtdt ,

where Wt , t ≥ 0 is a standard n-dimensional Brownian motion.


Integration-by-parts formula

Let F ∈ D1,2 with E(F ) = 0. Let f be a differentiable function withbounded derivative. Using that

F = LL−1F = −δ(DL−1F )

yields

E [f (F )F ] = −E [f (F )δ(DL−1F )]

= −E [〈D(f (F )),DL−1F 〉H ]

= E [f ′(F )〈DF ,−DL−1F 〉H ].

If F ∈ Hq, with q ≥ 1, then DL−1F = − 1q DF and

E [f (F )F ] =1q

E [f ′(F )‖DF‖2H ].


Define for almost all x in the support of F

gF (x) = E [〈DF ,−DL−1F 〉H |F = x ]

For any f ∈ C1b(R).

E [f (F )F ] = E [f ′(F )gF (F )]

Moreover, gF (F ) ≥ 0 almost surely. Indeed, takingf (x) =

∫ x0 ϕ(y)dy , where ϕ is smooth and non-negative we obtain

E [[〈DF ,−DL−1F 〉H |F ]ϕ(F )] ≥ 0,

because xf (x) ≥ 0.


an introduction to malliavin calculus and its applications lecture ii

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