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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.
David Nualart (Kansas University) May 2014 2 / 22
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 ).
David Nualart (Kansas University) May 2014 3 / 22
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).
David Nualart (Kansas University) May 2014 4 / 22
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).
David Nualart (Kansas University) May 2014 5 / 22
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].
David Nualart (Kansas University) May 2014 7 / 22
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 )
David Nualart (Kansas University) May 2014 8 / 22
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 .
David Nualart (Kansas University) May 2014 9 / 22
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).
David Nualart (Kansas University) May 2014 10 / 22
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.
David Nualart (Kansas University) May 2014 11 / 22
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 ′.
David Nualart (Kansas University) May 2014 12 / 22
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).
David Nualart (Kansas University) May 2014 13 / 22
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.
David Nualart (Kansas University) May 2014 14 / 22
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).
David Nualart (Kansas University) May 2014 15 / 22
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.
David Nualart (Kansas University) May 2014 16 / 22
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.
David Nualart (Kansas University) May 2014 17 / 22
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 .
David Nualart (Kansas University) May 2014 18 / 22
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 .
David Nualart (Kansas University) May 2014 19 / 22
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.
David Nualart (Kansas University) May 2014 20 / 22
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 ].
David Nualart (Kansas University) May 2014 21 / 22
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.
David Nualart (Kansas University) May 2014 22 / 22