An Introduction to Malliavin calculus and its applications Lecture 4: Density formulas

David Nualart Department of Mathematics Kansas University

University of Wyoming Summer School 2014

David Nualart (Kansas University)

May 2014

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First formula for probability densities

W = {Wt , t ∈ [0, T ]} is a Brownian motion on the canonical probability space (Ω, F, P). D and δ are the fundamental operators in the Malliavin calculus.

Proposition Let F be a random variable in the space D1,2 . Suppose that 2

DF kDF k2H

belongs to

the domain of the operator δ in L (Ω). Then the law of F has a continuous and bounded density given by    DF p(x) = E 1{F >x} δ . (1) kDF k2H

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Proof : Let ψ R ybe a nonnegative smooth function with compact support, and set ϕ(y ) = −∞ ψ(z)dz. Then ϕ(F ) belongs to D1,2 , and we can write hD(ϕ(F )), DF iH = ψ(F )kDF k2H . Using the duality formula we obtain "  #    DF DF = E ϕ(F )δ . E[ψ(F )] = E D(ϕ(F )) , kDF k2H H kDF k2H

(2)

By an approximation argument, Equation (2) holds for ψ(y ) = 1[a,b] (y), where a < b. As a consequence, we apply Fubini’s theorem to get !  " Z # F DF ψ(x)dx δ P(a ≤ F ≤ b) = E kDF k2H −∞   Z b  DF = E 1{F >x} δ dx kDF k2H a which implies the desired result. David Nualart (Kansas University)

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Equation (1) still holds under the hypotheses F ∈ D1,p and 0 DF ∈ D1,p (H) for some p, p0 > 1. kDF k2 H

Sufficient conditions are F ∈ D2,α and E(kDF k−2β ) < ∞ with 1 1 α + β < 1. Example : Let F = W (h). Then, DF = h and ! DF δ = W (h)khk−2 H . kDF k2H As a consequence, formula (1) yields   p(x) = khk−2 H E 1{F >x} F , which is true because p(x) is the density of N(0, khk2H ).

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Density estimates ¨ Fix p and q such that p−1 + q −1 = 1. By Holder’s inequality, and using (1) with P(F < x), we obtain

 

DF 1/q

, (3) p(x) ≤ (P(|F | > |x|)) δ

2 kDF kH p for all x ∈ R. Then, using Meyer’s inequalities we dan deduce the following result.

Proposition Let q, α, β be three positive real numbers such that q −1 + α−1 + β −1 = 1. Let −2β F be a random variable in the space D2,α , such that E(kDF kH ) < ∞. Then the density p(x) of F can be estimated as follows 1/q

p(x) ≤ cq,α,β (P(|F | > |x|)) 



−1 −2 × E(kDF kH ) + D 2 F Lα (Ω;H⊗H) kDF kH .

(4)

β

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Example I Rt

u(s)dWs , where u = {u(t), t ∈ [0, T ]} is an adapted process such  R T that |u(t)| ≥ ρ > 0 for some constant ρ, E 0 u(t)2 dt < ∞, u(t) ∈ D2,2 for Set Mt =

0

each t ∈ [0, T ], and  λ := sup E(|Ds ut |p ) + sup E  s,t∈[0,T ]

r ,s∈[0,T ]

Z

!p/2 

T

|Dr2,s ut |p dt

 < ∞,

0

for some p > 3. Then, appplying Proposition 2 one can show that the density of Mt , denoted by pt (x) satisfies 1 c pt (x) ≤ √ P(|Mt | > |x|) q , t

for all t > 0, where q >

David Nualart (Kansas University)

p p−3

and the constant c depends on λ, ρ and p.

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Example II Consider the process ξ = {ξr ,t , 0 ≤ r ≤ t ≤ T } solution of the stochastic differential equation Z tZ ξr ,t = x + Wt − Wr +

h (y − ξr ,u ) Z (du, dy) , r

R

where x ∈ R. W is a standard Brownian motion on R. Z is a Brownian sheet independent of W . h ∈ H22 (R). (i) ξ represents the position of a particle in a random environment. (ii) A measure valued process obtained as the limit of critical branching particles moving with the dynamics of ξ was studied by Dawson, Li and Wang ’01. David Nualart (Kansas University)

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For r ≤ t we define the conditional transition density given Z by pW (r , x; t, y) = P W (ξt ∈ dy |ξr = x) Using the formula   pW (r , x; t, y) = E W 1{ξr ,t >y} δ(ur ,t ) , one can derive the following result :

Lemma 2

Let c = 1 ∨ khk2 . For any 0 ≤ r < t ≤ T , y ∈ R and p ≥ 1, ! 2 1 (x − y ) W kp (r , x; t, y ) k2p ≤ 2Kp exp − (t − r )− 2 , 64pc (t − r ) where Kp is a constant depending on p. ¨ This lemma has been used by Fei, Hu and Nualart ’11 to establish the Holder continuity in space and time of the solution of the SPDE satisfied by the density of the associated limiting branching particles. David Nualart (Kansas University)

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Second formula for densities

Let F ∈ D1,2 be such that E[F ] = 0. Define gF (x) = E[hDF , −DL−1 F iH |F = x] ≥ 0.

Theorem (Nourdin Viens 09’) The law of F has a density p if and only if gF (F ) > 0 a.s. In this case the support of p is a closed interval containing zero and for all x in the support of p  Z x  ydy E[|F |] p(x) = exp − . 2gF (x) 0 gF (y )

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Proof : Suppose that F has a density p and its support is R. Let φ be a smooth function with compact support and Φ0 = φ. Then Z E[φ(F )gF (F )] = E[Φ(F )F ] = Φ(x)xp(x)dx R Z = φ(x)ϕ(x)dx, R

where ϕ(x) =

R∞ x

yp(y )dy . This implies ϕ(x) = p(x)gF (x).

Taking into account that ϕ0 (x) = −xp(x) we obtain ϕ0 (x) x =− . ϕ(x) gF (x) Using that ϕ(0) = 12 E[|F |], we can write  Z x  1 ydy . ϕ(x) = E[|F |] exp − 2 0 gF (y ) which completes the proof. David Nualart (Kansas University)

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Corollary 2 2 , σmax > 0 such that If there exist σmin 2 2 σmin ≤ gF (F ) ≤ σmax

a.s., then F has a density p satisfying     E[F ] E[F ] x2 x2 ≤ p(x) ≤ exp − 2 exp − 2 . 2 2 2σmax 2σmax 2σmin 2σmin Using Mehler’s formula and −DL−1 F = Z gF (F ) =

∞

R∞ 0

e−t Tt (DF )dt, we obtain

h i p e−t E hDF , E 0 [DF (e−t W + 1 − e−2t W 0 )]iH |F dt.

0

The above corollary can be applied √ if we have uniform upper and lower bounds for hDF , E 0 [DF (e−t W + 1 − e−2t W 0 )]iH .

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Example F = max BtH − E t∈[a,b]



 max BtH ,

t∈[a,b]

where 0 < a < b ≤ T and B H = {BtH , t ∈ [0, T ]} is a fractional Brownian motion with Hurst parameter H ∈ [1/2, 1). That is B H is a zero mean Gaussian process with covariance E[BtH BsH ] =


1 2H t + s2H − |t − s|2H . 2

Then, the previous corollary holds with σmin = aH and σmax = bH . The Malliavin calculus can be developed replacing H by the closed span of the indicator functions under the inner product h1[0,t] , 1[0,t] iH = E[BtH BsH ]. F ∈ D1,2 and Dr F = 1[0,τ ] (r ), where τ is the random point in [0, T ] where B H attains its maximum. The lower and upper bounds for gF follow from a2H ≤ E[BtH BsH ] ≤ b2H . David Nualart (Kansas University)

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Absolute continuity of random vectors Let F = (F 1 , . . . , F m ) be such that Fi ∈ D1,2 for i = 1, . . . , m. We define the Malliavin matrix of F as the random symmetric nonnegative definite matrix γF = (hDF i , DF j iH )1≤i,j≤m .

Theorem (Bouleau Hirsch ’91) If det γF > 0 a.s., then the law of F is absolutely continuous with respect to the Lebesgue measure on Rm . The proof of this theorem is based on the coarea formula and uses the techniques of geometric measure theory. The measure (det γF ◦ P) ◦ F −1 is always absolutely continuous, that is, P (F ∈ B , det γF > 0) = 0 m

for any B ∈ B(R ) of zero Lebesgue measure. 2

In the one-dimensional case, γF = kDF k . David Nualart (Kansas University)

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Smoothness of the density of random vectors Definition Let F = (F 1 , . . . , F m ) be a random vector such that Fi ∈ D1,2 for i = 1, . . . , m. We say that F is nondegenerate if E[(det γF )−p ] < ∞ for all p ≥ 2. Set ∂i = ∂/∂xi , and for any multiindex α ∈ {1, . . . , m}k , k ≥ 1, we denote by ∂α the partial derivative ∂ k /(∂xα1 · · · ∂xαk ).

Lemma Let γ be an m × m random matrix such that γ ij ∈ D1,∞ for all i, j and
ij E[(det γ)−p ] < ∞ for all p ≥ 2. Then γ −1 belongs to D1,∞ for all i, j, and D γ −1


ij

=−

m X

γ −1


ik

γ −1


lj

Dγ kl .

k ,l=1

Proof : Approximate γ −1 by γ−1 = (det γ + )−1 A(γ), where A(γ) is the adjoint matrix of γ. David Nualart (Kansas University)

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Proposition (Integration by parts) Let F = (F 1 , . . . , F m ) be a nondegenerate random vector. Fix k ≥ 1 and suppose that Fi ∈ Dk+1,∞ for i = 1, . . . , m. Let G ∈ D∞ and let ϕ ∈ Cp∞ (Rm ). Then for any multiindex α ∈ {1, . . . , m}k , there exists an element Hα (F , G) ∈ D∞ such that E [∂α ϕ(F )G] = E [ϕ(F )Hα (F , G)] , where the elements Hα (F , G) are recursively given by H(i) (F , G)

=

m     X δ G γF−1 ij DF j ,

(5)

j=1

Hα (F , G)

= Hαk (F , H(α1 ,...,αk −1 ) (F , G)).

(6)

For any p > 1 there exist constants β, γ > 1 and integers n, m such that

m

n kHα (F , G)kp ≤ cp,q det γF−1 kDF kk,γ kGkk,q . β

David Nualart (Kansas University)

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Proof : By the chain rule we have hD(ϕ(F )), DF j iH =

m X

∂i ϕ(F )hDF i , DF j iH =

i=1

m X

∂i ϕ(F )γFij ,

i=1

and, consequently, ∂i ϕ(F ) =

m X

hD(ϕ(F )), DF j iH (γF−1 )ji .

j=1

Taking expectations and using the duality relationship yields   E [∂i ϕ(F )G] = E ϕ(F )H(i) (F , G) ,    ij Pm −1 j where H(i) = j=1 δ G γF DF . Notice that the continuity of the operator δ and the previous lemma imply that H(i) belongs to Lp for any p ≥ 2. We can finish the proof by a recurrence argument.

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Density formula Proposition Let F = (F 1 , . . . , F m ) be a nondegenerate random vector such that Fi ∈ Dm+1,∞ for i = 1, . . . , m. Then, F has a continuous and bounded density given by   p(x) = E 1{F >x} Hα (F , 1) . where α = (1, 2, . . . , m). Recall that Hα (F , 1) =

m X

     δ (γF−1 )1j1 DF j1 δ (γF−1 )2j2 DF j2 . . . δ (γF−1 )mjm DF jm . . . .

j1 ,...,jm =1

David Nualart (Kansas University)

May 2014

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Proof : Equality (6) applied to the multiindex α = (1, 2, . . . , m) yields E [∂α ϕ(F )] = E[ϕ(F )Hα (F , 1)]. Notice that Z

F1

Z

Fm

···

ϕ(F ) = −∞

∂α ϕ(x)dx. −∞

Hence, by Fubini’s theorem we can write Z   E [∂α ϕ(F )] = ∂α ϕ(x)E 1{F >x} Hα (F , 1) dx.

(7)

Rm

We can take as ∂α ϕ any function in C0∞ (Rm ) and the result follows.

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Theorem (Criterion for smoothness of densities) Let F = (F 1 , . . . , F m ) be a nondegenerate random vector such that F i ∈ D∞ for all i = 1, . . . , m. Then the law of F possesses an infinitely differentiable density. Proof : For any multiindex β we have (with α = (1, 2, . . . , m)) E [∂β ∂α ϕ(F )] = =

E[ϕ(F )Hβ (F , Hα (F , 1)))] Z   ∂α ϕ(x)E 1{F >x} Hβ (F , Hα (F , 1)) dx. Rm

Hence, for any ξ ∈ C0∞ (Rm ) Z Z ∂β ξ(x)p(x)dx = Rm

  ξ(x)E 1{F >x} Hβ (F , Hα (F , 1)) dx.

Rm

Therefore p(x) is infinitely differentiable, and for any multiindex β we have   ∂β p(x) = (−1)|β| E 1{F >x} Hβ (F , (Hα (F , 1)) .

David Nualart (Kansas University)

May 2014

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Density formulas using the Riesz transform Let Qm be the fundamental solution to the Laplace equation ∆Qm = δ0 on Rm , m ≥ 2. That is, 1 −1 Q2 (x) = a2−1 ln , Qm (x) = am |x|2−m , m > 2, |x| where am is the area of the unit sphere in Rm . We know that

xi , |x|m where cm = 2(m − 2)/am if m > 2 and c2 = 2/a2 . ∂i Qm (x) = −cm

Lemma Any function ϕ in C01 (Rm ) can be written as ϕ(x) = ∇ϕ ∗ ∇Qm (x). Proof : ∇ϕ ∗ ∇Qm (x) = ϕ ∗ ∆Qm (x) = ϕ(x). David Nualart (Kansas University)

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Theorem (Bally-Caramellino ’11, ’12) Let F be an m-dimensional non-degenerate random vector whose components are in D2,∞ . Then the law of F admits a continuous and bounded density p given by p(x) =

m X


E ∂i Qm (F − x)H(i) (F , 1) .

i=1

Recall that

m   X H(i) (F , 1) = δ (γF−1 )ij DF j . j=1

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Proof : Let ϕ ∈ C01 (Rm ). Applying the previous lemma, Fubini’s theorem, and the integration by parts formula we get m Z X E (ϕ(F )) = ∂i Qm (y )E (∂i ϕ(F − y )) dy =

m i=1 R Z m X

i=1

=

m Z X i=1


∂i Qm (y )E ϕ(F − y )H(i) (F , 1) dy

Rm


ϕ(y )E ∂i Qm (F − y)H(i) (F , 1) dy .

Rm

The use of Fubini’s theorem needs to be justified by showing that all the functions are integrable. Assume that the support of ϕ is included in the ball BR (0) for some R > 1. Then, Z Z E |∂i Qm (y )∂i ϕ(F − y )| dy ≤ |∂i ϕ|∞ E |∂i Qm (y)|dy Rm

{y:|F |−R≤|y |≤|F |+R}

Z

|F |+R

≤ Cm |∂i ϕ|∞ E |F |−R

r m−1 r dr rm

= 2Cm R|∂i ϕ|∞ < ∞. David Nualart (Kansas University)

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Lemma (Stroock) For any p > m there exists a constant c depending only on m and p such that  m kpk∞ ≤ c max kH(i) (F , 1)kp . 1≤i≤m

Proof : Suppose m ≥ 3. From p(x) =

m X


E ∂i Qm (F − x)H(i) (F , 1)

i=1

¨ Holder’s inequality with

1 p

+

1 q

= 1 and the estimate

|∂i Qm (F − x)| ≤ cm |F − x|1−m yields  1/q p(x) ≤ mcm A E[|F − x|(1−m)q ] , where A = max1≤i≤m kH(i) (F , 1)kp . David Nualart (Kansas University)

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i) Suppose first that p is bounded by a constant M. We can write for any  > 0, Z (1−m)q (1−N)q E[|F − x| ] ≤  + |z − x|(1−N)q p(x)dx |z−x|≤

≤ 

(1−N)q

p−N

+ CN  p−1 M.

Therefore,  M ≤ ANkN

1−N



1 q

+ CN 

p−N p

M

1 q

 .

Now we minimize with respect to  and we obtain 1

M ≤ ACN,p M 1− N N for some constant cN,p , which implies M ≤ cN,p AN .

ii) If p is not bounded, we apply the procedure to p ∗ ψδ , where ψδ is an approximation of the identity and let δ tend to zero at the end.

David Nualart (Kansas University)

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