The determinant of the iterated Malliavin matrix and the density of a couple of multiple integrals

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The determinant of the iterated Malliavin matrix and the density of a couple of multiple integrals

David Nualart, Ciprian Tudor

To cite this version:

David Nualart, Ciprian Tudor. The determinant of the iterated Malliavin matrix and the density of

a couple of multiple integrals. 2014. �hal-00949119�

The determinant of the iterated Malliavin matrix and the density of a couple of multiple integrals

David Nualart

Department of Mathematics University of Kansas Lawrence, Kansas 66045, USA

nualart@ku.edu

Ciprian A. Tudor

Laboratoire Paul Painlev´e

Universit´e de Lille I

F-59655 Villeneuve d’Ascq, France

and Academy of Economical Studies, Bucharest, Romania tudor@math.univ-lille1.fr

Abstract

The aim of this paper is to show an estimate for the determinant of the covariance of a two-dimensional vector of multiple stochastic integrals of the same order in terms of a linear combination of the expectation of the deter- minant of its iterated Malliavin matrices. As an application we show that the vector is absolutely continuous if and only if its components are proportional.

2010 AMS Classification Numbers: 60F05, 60H05, 91G70.

Key words: multiple stochastic integrals, Wiener chaos, iterated Malli- avin matrix, covariance matrix, existence of density.

∗Supported by the NSF grant DMS-1208625.

†Supported by the CNCS grant PN-II-ID-PCCE-2011-2-0015.

1 Introduction

A basic result in Malliavin calculus says that if the Malliavin matrix Λ = ( h DF

, DF

i

)

of a d-dimensional random vector F = (F

, . . . , F

) is nonsingular almost surely, then this vector has an absolutely continuous law with respect to the Lebesgue measure in

. In the special case of vectors whose components belong to a finite sum of Wiener chaos, Nourdin, Nualart and Poly proved in [1] that the following conditions are equivalent:

(a) The law of F is not absolutely continuous.

(b) E det Λ = 0.

A natural question is the relation between E det Λ and the determinant of the covariance matrix C of the random vector F. Clearly if det C = 0, then the components of F are linearly dependent and the law of F is not absolutely continuous, which implies E det Λ = 0. The converse is not true if d ≥ 3. For instance, the vector (F

, F

, F

F

), where F

and F

are two non-zero indepen- dent random variables in the first chaos, satisfies det Λ = 0 but det C 6 = 0.

The purpose of this paper is to show the equivalence between E det Λ = 0 and det C = 0 in the particular case of a two-dimensional random vector (F, G) whose components are multiple stochastic integrals of the same order n.

This implies that the random vector (F, G) has an absolutely continuous law with respect to the Lebesgue measure on

if and only if its components are proportional, as in the Gaussian case. This result was established for n = 2 in [1], and for n = 3, 4 in [6]. Our proof in the general case n ≥ 2 is based on the notion of iterated Malliavin matrix and the computation of the expectation of its determinant.

In connection with this equivalence we will derive an inequality relating E det Λ and det C, which has its own interest. In the case of double stochastic integrals, that is if n = 2, it was proved in [1] that

E det Λ ≥ 4 det C.

We extend this inequality proving that

E det Λ ≥ c

det C

holds for n = 3, 4 with c

=

and c

=

. For n ≥ 5 we obtain a more involved inequality, where in the left hand side we have a linear combination (with positive coefficients) of the expectation of the iterated Mallavin matrices of (F, G) of order k for 1 ≤ k ≤

2

(see Theorem 2 below).

The paper is organized as follows. In Section 2 we present some preliminary

results and notation. Section 3 contains a general decomposition of the deter-

minant of the iterated Malliavin matrix of a two-dimensional random vector

into a sum of squares. In Section 4 we prove our main result which is based on a further decomposition of the determinant of the iterated Malliavin matrix of a vector whose components are multiple stochastic integrals. Finally, the ap- plication to the characterization of absolutely continuity is obtained in Section 5.

2 Preliminaries

We briefly describe the tools from the analysis on Wiener space that we will need in our work. For complete presentations, we refer to [5] or [3]. Let H be a real and separable Hilbert space and consider an isonormal process (W (h), h ∈ H).

That is, (W (h), h ∈ H) is a Gaussian family of centered random variables on a probability space (Ω, F , P ) such that EW (h)W (g) = h f, g i

for every h, g ∈ H.

Assume that the σ-algebra F is generated by W .

For any integer n ≥ 1 we denote by H

the nth Wiener chaos generated by W . That is, H

is the vector subspace of L

(Ω) generated by the random variables (H

(W (h)), h ∈ H, k h k

= 1) where H

the Hermite polynomial of degree n. We denote by H

the space of constant random variables. Let H

and H

denote, respectively, the nth tensor product and the nth symmetric tensor product of H. For any n ≥ 1, the mapping I

(h

) = H

(W (h)) can be extended to an isometry between the symmetric tensor product space H

endowed with the norm √

n! k · k

and the nth Wiener chaos H

. For any f ∈ H

, the random variable I

(f ) is called the multiple Wiener Itˆ o integral of f with respect to W .

Consider (e

)

a complete orthonormal system in H and let f ∈ H

, g ∈ H

be two symmetric tensors with n, m ≥ 1. Then

f =

j1,...,jn≥1

f

e

⊗ · · · ⊗ e

(1)

and

g =

k₁,...,km≥1

g

e

⊗ · · · ⊗ e

, (2) where the coefficients are given by f

= h f, e

⊗ · · · ⊗ e

i and g

= h g, e

⊗ · · · ⊗ e

i . These coefficients are symmetric, that is, they satisfy f

= f

and g

= g

for every permutation σ of the set { 1, . . . , n } and for every permutation π of the set { 1, . . . , m } .

Note that, throughout the paper we will usually omit the subindex H

in the notation for the norm and the scalar product in H

for any k ≥ 1.

If f ∈ H

, g ∈ H

are symmetric tensors given by (1) and (2), respec-

tively, then the contraction of order r of f and g is given by f ⊗

g =

i1,...,ir≥1

X

j1,...,jn−r≥1

X

k1,...,km−r≥1

f

g

× e

⊗ · · · ⊗ e

⊗ e

⊗ · · · ⊗ e

(3) for every r = 0, . . . , m ∧ n. In particular f ⊗

g = f ⊗ g. Note that f ⊗

g belongs to H

for every r = 0, . . . , m ∧ n and it is not in general symmetric.

We will denote by f ⊗ ˜

g the symmetrization of f ⊗

g. In the particular case when H = L

(T, B , µ) where µ is a sigma-finite measure without atoms, (3) becomes

(f ⊗

g)(t

, . . . , t

) =

T^r

dµ(u

) · · · dµ(u

)f(u

, . . . , u

, t

, . . . , t

)

× g(u

, . . . , u

, t

, . . . , t

). (4) An important role will be played by the following product formula for mul- tiple Wiener-Itˆ o integrals: if f ∈ H

, g ∈ H

are symmetric tensors, then

I

(f )I

(g) =

m∧nX

r=0

r!C

C

I

f ⊗ ˜

g

. (5)

We will need some elements of the Malliavin calculus with respect to the isonormal Gaussian process W . Let S be the set of all smooth and cylindrical random variables of the form

F = ϕ (W (h

), . . . , W (h

)) , (6) where n ≥ 1, ϕ :

→

is a infinitely differentiable function with compact support, and h

∈ H for i = 1, .., n. If F is given by (6), the Malliavin derivative of F with respect to W is the element of L

(Ω; H) defined as

DF =

i=1

∂ϕ

∂x

(W (h

), . . . , W (h

)) h

.

By iteration, one can define the kth derivative D

F for every k ≥ 2, which is an element of L

(Ω; H

). For k ≥ 1,

denotes the closure of S with respect to the norm k · k

, defined by the relation

k F k

= E

| F |

+

Xk i=1

E

k D

F k

.

If F = I

(f ), where f ∈ H

and I

(f ) denotes the multiple integral of order n with respect to W , then

DI

(f ) = n

I

(f ⊗

e

) e

.

More generally, for any 1 ≤ k ≤ n, the iterated Malliavin derivative of I

(f ) is given by

D

I

(f ) = n!

(n − k)!

X

j1,...,jk≥1

I

(f

) e

⊗ · · · ⊗ e

, where

f

= f ⊗

(e

⊗ · · · ⊗ e

). (7) We denote by δ the adjoint of the operator D, also called the divergence operator or Skorohod integral. A random element u ∈ L

(Ω; H) belongs to the domain of δ, denoted Domδ, if and only if it verifies

| E h DF, u i

| ≤ c

E(F

)

for any F ∈

, where c

is a constant depending only on u. If u ∈ Domδ, then the random variable δ(u) is defined by the duality relationship

E(F δ(u)) = E h DF, u i

,

which holds for every F ∈

. If F = I

(f ) is a multiple stochastic integral of order n, with f ∈ H

, then DF belongs to the domain of δ and

δDF = nF. (8)

3 Decomposition of the determinant of the iterated Malliavin matrix

In this section we obtain a decomposition into a sum of a squares for the determinant of the iterated Malliavin matrix of a 2-dimensional random vector.

We recall that if F, G are two random variables in the space

, the Malliavin matrix of the random vector (F, G) is the defined as the following 2 × 2 random matrix

Λ =

k DF k

h DF, DG i

h DF, DG i

k DF k

.

More generally, fix k ≥ 2 and suppose that F, G are two random variables in

. The kth iterated Malliavin matrix of the vector (F, G) is defined as

Λ

=

k D

F k

h D

F, D

G i

h D

F, D

G i

k D

G k

,

We set Λ

= Λ. For every j

, . . . , j

≥ 1, we will write D

k

F = h D

F, e

⊗ · · · ⊗ e

i

.

The next proposition provides an expression of the determinant of the iterated

Malliavin matrix of a random vector as a sum of squared random variables.

Proposition 1 Suppose that (F, G) is a 2-dimensional random vector whose components belong to

for some k ≥ 1. Let Λ

be the kth iterated Malliavin matrix of (F, G). Then

det Λ

= 1 2

X

i1,...,ik,l1,...,lk≥1

D

k

F D

1,...,l_k

G − D

1,...,l_k

F D

k

G

. (9) Proof: For every k ≥ 1 we have

k D

F k

=

i₁,...,i_k≥1

D

1,...,i_k

F

,

k D

G k

=

i₁,...,i_k≥1

D

1,...,ik

G

and

h D

F, D

G i

=

i1,...,ik≥1

D

k

F D

k

G.

Thus

det Λ

=

i₁,...,i_k≥1

D

1,...,i_k

F

i₁,...,i_k≥1

D

1,...,i_k

G

−



 X

i1,...,ik≥1

D

1,...,ik

F D

1,...,ik

G





2

= 1

2

X

i1,..,ik,l1,..,lk≥1

D

1,...,ik

F D

1,...,lk

G − D

1,...,lk

F D

1,...,ik

G

.

4 The iterated Malliavin matrix of a two- dimensional vector of multiple integrals

Throughout this section, we assume that the components of the random vector (F, G) are multiple Wiener-Itˆ o integrals. More precisely, we will fix n, m ≥ 1 and we will consider the vector

(F, G) = (I

(f ), I

(g))

where f ∈ H

and g ∈ H

. Since for every 1 ≤ k ≤ min(n, m), D

1,...,i_k

F = n!

(n − k)! I

(f

)

(with f

defined by (7)) and D

k

G = m!

(m − k)! I

(g

) formula (9) reduces to

det Λ

= 1 2

n!

(n − k)!

m!

(m − k)!

2 X

i1,...,ik,l1,...,lk≥1

[I

(f

) I

(g

)

− I

(f

) I

(g

)]

.

By the product formula for multiple integrals (5) we can write det Λ

= 1

2

n!

(n − k)!

m!

(m − k)!

2 X

i1,...,ik,l1,...,lk≥1





(n−k)∧(m−k)X

r=0

r!C

C

I

[f

⊗

g

− f

⊗

g

]





2

.

Taking the mathematical expectation, the isometry of multiple integrals implies that

E det Λ

= 1 2

n!

(n − k)!

m!

(m − k)!

2 X

i₁,...,i_k,l₁,...,l_k≥1 (n−k)∧(m−k)X

r=0

r!C

C

(m + n − 2k − 2r)!

×k f

⊗ ˜

g

− f

⊗ ˜

g

k

:=

(n−k)∧(m−k)X

r=0

T

,

where T

= 1

2 α

i1,...,ik,l1,...,lk≥1

k f

⊗ ˜

g

− f

⊗ ˜

g

k

, (10) with

α

=

n!m!

(n − k − r)!(m − k − r)!r!

2

(m + n − 2k − 2r)!.

We will explicitly compute the terms T

in (10). To do this, we will need

several auxiliary lemmas. The first one is an immediate consequence of the

definition of contraction.

Lemma 1 Let f ∈ H

, g ∈ H

. Then for every k, r ≥ 0 such that k + r ≤

m ∧ n,

i1,...,ik≥1

f

⊗

g

= f ⊗

g.

The next lemma summarizes the results in Lemmas 3 and 4 in [6] (see also Lemma 2.2 in [4]).

Lemma 2 Assume f, h ∈ H

and g, ℓ ∈ H

. (i) For every r = 0, . . . , (m − 1) ∧ (n − 1) we have

h f ⊗

h, g ⊗

ℓ i = h f ⊗

g, h ⊗

ℓ i . (ii) The following equality holds

h f ⊗ ˜ g, ℓ ⊗ ˜ h i = m!n!

(m + n)!

m∧nX

r=0

C

C

h f ⊗

ℓ, h ⊗

g i .

We are now ready to calculate the term T

.

Proposition 2 Let f ∈ H

, g ∈ H

. Let T

be given by (10). Then for every 1 ≤ k ≤ min(m, n)

T

= m!

n!

(m − k)!(n − k)!

(m−k)∧(n−k)X

s=0

C

C

k f ⊗

g k

− k f ⊗

g k

.

Proof: From (10) we can write T

= 1

2 α

i1,...,ik,l1,...,lk≥1

k f

⊗ ˜ g

− f

⊗ ˜ g

k

= α

i₁,...,i_k,l₁,...,l_k≥1

h

k f

⊗ ˜ g

k

−h f

⊗ ˜ g

, g

⊗ ˜ f

i

. (11)

By Lemma 2, point (ii) and point (i)

k f

⊗ ˜ g

k

= h f

⊗ ˜ g

, f

⊗ ˜ g

) i

= (m − k)!(n − k)!

(m + n − 2k)!

(m−k)∧(n−k)X

s=0

C

C

×h f

⊗

g

, f

⊗

g

i

= (m − k)!(n − k)!

(m + n − 2k)!

(m−k)∧(n−k)X

s=0

C

C

×h f

⊗

f

, g

⊗

g

i . (12)

Also, Lemma 1 and Lemma 2 point (i) imply

i₁,...,i_k,l₁,...,l_k≥1

h f

⊗

f

, g

⊗

g

i

= h f ⊗

f, g ⊗

g i = k f ⊗

g k

. (13) On the other hand, using again Lemma 2, point (ii)

h f

⊗ ˜ g

, g

) ˜ ⊗ f

i = (m − k)!(n − k)!

(m + n − 2k)!

×

(m−k)∧(n−k)X

s=0

C

C

h f

⊗

g

, f

⊗

g

i . (14) Again, Lemma 1 and Lemma 2 point (i) imply

X

i1,...,ik,l1,...,lk≥1

h f

⊗

g

, f

⊗

g

i

= h f ⊗

g, f ⊗

g i = k f ⊗

g k

. (15) Then, substituting (12), (13), (14) and (15) into (11) yields the desired result.

It is also possible to compute the terms T

for every 1 ≤ r ≤ (n − k) ∧ (m − k) but the corresponding expressions are more complicated, involving some kind of contractions of contractions. In order to obtain this type of formula we need the following generalization of point (ii) in Lemma 2.

For f, h ∈ H

and g, ℓ ∈ H

and for r, s ≥ 0 such that r + s ≤ m ∧ n we denote by (f ⊗

g) ⊗

(ℓ ⊗

h) the contraction of r coordinates between f and g and between ℓ and h, s coordinates between f and ℓ and between g and h, n − r − s coordinates between f and h and m − r − s coordinates between g an ℓ. That is,

(f ⊗

g) ⊗

(ℓ ⊗

h) =

h f

ih g

i

×h ℓ

ih h

i , where the sum runs over all indices greater or equal than one. Notice that

(f ⊗

g) ⊗

(ℓ ⊗

h) = (f ⊗

ℓ) ⊗

(g ⊗

h).

Lemma 3 Assume f, h ∈ H

and g, ℓ ∈ H

. Then for every r = 0, . . . , (m − 1) ∧ (n − 1) we have

h f ⊗ ˜

g, ℓ ⊗ ˜

h i = (n − r)!(m − r)!

(m + n − 2r)!

(m−r)∧(n−r)X

s=0

C

C

(f ⊗

g) ⊗

(ℓ ⊗

h).

Proof: We can write

f ⊗ ˜

g =

i₁,...,ir

f

⊗ ˜ g

and

ℓ ⊗ ˜

h =

i1,...,ir

ℓ

⊗ ˜ h

. Then, Lemma 2 point (ii) gives

h f ⊗ ˜

g, ℓ ⊗ ˜

h i =

i1,...,ir,l1,...lr

h f

⊗ ˜ g

, ℓ

⊗ ˜ h

i

= (n − r)!(m − r)!

(m + n − 2r)!

(m−r)∧(n−r)X

s=0

C

C

h f

⊗

ℓ

, h

⊗

g

i , which implies the desired result.

Notice that for r = 0,

(f ⊗ g) ⊗

(ℓ ⊗ h) = h f ⊗

ℓ, h ⊗

g i ,

so Lemma 2 point (ii) is a particular case of Lemma 3 when r = 0.

Proposition 3 Let (F, G) = (I

(f ), I

(g)) with f ∈ H

and g ∈ H

. Then, for every r = 1, . . . , (n − k) ∧ (m − k)

T

= β

(n−k−r)∧(m−k−r)

X

s=0

C

C

× (f ⊗

g) ⊗

(g ⊗

f ) − (f ⊗

g) ⊗

(g ⊗

f )

, (16) where

β

= n!

m!

(n − k − r)!(m − k − r)!(r!)

. Proof: From (10) we can write

T

= α

i1,...,ik,l1,...,lk≥1

k f

⊗ ˜

g

) k

−h f

⊗ ˜

g

, f

) ˜ ⊗

g

i

. (17)

Applying Lemma 3 yields

k f

⊗ ˜

g

k

h f

⊗ ˜

g

, f

) ˜ ⊗

g

i

= (n − k − r)!(m − k − r)!

(m + n − 2k − 2r)!

(n−k−r)∧(m−k−r)X

s=0

C

C

× (f

⊗

g

) ⊗

(g

⊗

f

) . (18)

Notice that

i1,...,ik,l1,...,lk≥1

(f

⊗

g

) ⊗

(g

⊗

f

) = (f ⊗

g) ⊗

(g ⊗

f ).

(19) Analogously, we get

h f

⊗ ˜

g

, f

⊗ ˜

g

i

= (n − k − r)!(m − k − r)!

(m + n − 2k − 2r)!

(n−k−r)∧(m−k−r)X

s=0

C

C

× (f

⊗

g

) ⊗

(g

) ⊗

f

) , (20)

and

i1,...,ik,l1,...,lk≥1

(f

⊗

g

) ⊗

(g

⊗

f

) = (f ⊗

g) ⊗

(g ⊗

f ).

(21) Substituting (18), (19), (20) and (21) into (17) we obtain the desired formula.

In the particular case n = m, the expression (16) can be written as T

=

n−k−rX

s=0

T

, (22)

where

T

= (n!)

((n − k − r)!r!)

(C

)

(f ⊗

g) ⊗

(g ⊗

f ) − (f ⊗

g) ⊗

(g ⊗

f ) . The last term in (22) obtained for r = n − k is given by the following expression.

Corollary 1 Let (F, G) = (I

(f ), I

(g)) with f, g ∈ H

. Then for k = 1, . . . , n − 1

T

= n!

(n − k)!

k f ⊗

g k

− h f ⊗

g, g ⊗

f i .

Proof: When r = n − k, there is only one terms in the sum (22), obtained for s = 0. It is easy to see that,

(f ⊗

g) ⊗

(g ⊗

f ) = (f ⊗ g) ⊗

(g ⊗ f ) = k f ⊗

g k

and

(f ⊗

g) ⊗

(g ⊗

) = h f ⊗

g, g ⊗

f i .

We obtain the following expression for the determinant of the kth Malliavin

matrix.

Theorem 1 Let f ∈ H

, g ∈ H

. Then for every 1 ≤ k ≤ m ∧ n, E det Λ

= m!

n!

(m − k)!(n − k)!

(m−k)∧(n−k)

X

s=0

C

C

×

k f ⊗

g k

− k f ⊗

g k

+ R

, where R

=

r=1

T

and T

is given by (16).

In the case of multiple integrals of the same order (i.e. m = n) we have the following result.

Corollary 2 If f, g ∈ H

, the determinant of the kth iterated Malliavin ma- trix of (F, G) = (I

(f ), I

(g)) can be written as

E det Λ

= n!

(n − k)!

n−kX

s=0

(C

)

k f ⊗

g k

− k f ⊗

g k

+ R

. Example 1 Suppose m = n = 3 and k = 2. Then

E det Λ

= (3!)

k f ⊗

g k

− k f ⊗

g k

+ k f ⊗

g k

− k f ⊗

g k

+ R

. Suppose m = n = 4 and k = 2. Then

E det Λ

= (4!)

2!

k f ⊗

g k

− k f ⊗

g k

+ 4( k f ⊗

g k

− k f ⊗

g k

)

+ R

. Our next objective is to relate the expectation of the iterated Malliavin matrix E det Λ

with the covariance matrix of the vector (F, G) in the case n = m. We recall that

det C = n!

[ k f k

k g k

− h f, g i

].

Theorem 2 For any f, g ∈ H

, if F = I

(f) and G = I

(g), we have [

]

X

s=2

n(n − 2s)

s!

E det Λ

+ (n − 1)

E det Λ

≥ n

det C.

Proof: From Corollary 2, taking into account that R

≥ 0, we can write E det Λ

≥ [nn!]

n−1X

s=0

(C

)

k f ⊗

g k

− k f ⊗

g k

= n

det C + [nn!]

n−1X

s=1

(C

)

− (C

)

k f ⊗

g k

.

Notice that (C

)

− (C

)

= − [(C

)

− (C

)

]. Therefore, we con- clude that

E det Λ

≥ n

det C + [nn!]

[

]

X

s=1

(C

)

− (C

)

× k f ⊗

g k

− k f ⊗

g k

= n

det C + [

]

X

s=1

γ

k f ⊗

g k

− k f ⊗

g k

, (23) where

γ

=

n!

(n − s)!s!

2

n(n − 2s).

Notice that γ

≥ 0 if s ≤

2

. We can write, using Lemma 2 point (i) and Corollary 1

k f ⊗

g k

− k f ⊗

g k

= k f ⊗

g k

− h f ⊗

g, g ⊗

f i

− k f ⊗

g k

− h f ⊗

g, g ⊗

f i

≥ − (n − s)!

n!

T

≥ − (n − s)!

n!

E det Λ

.(24) Substituting (24) into (23) yields

E det Λ

≥ n

det C − [

]

X

s=1

n(n − 2s)

s!

E det Λ

, which implies the desired result.

Remark 1 In the particular case n = 2 we obtain E det Λ

≥ 4 det C, which was proved in [1]. For n = 3 we get E det Λ

≥

det C, and for n = 4, E det Λ

≥

det C. Only if n ≥ 5 we need the expectation of the iterated Malliavin matrix to control the determinant of the covariance matrix.

5 The density of a couple of multiple inte- grals

In this section, we show that a random vector of dimension 2 whose components

are multiple integrals in the same Wiener chaos either admits a density with

respect to the Lebesque measure, or its components are proportional. We also

show that a necessary and sufficient condition for such a vector to not have a

density is that at least one of its iterated Malliavin matrices vanishes almost surely. In the sequel we fix a vector (F, G) = (I

(f ), I

(g)) with f, g ∈ H

.

In the following result we show that, if the determinant of an iterated Malli- avin matrix of a couple of multiple integrals vanishes, the determinant of the any other iterated Malliavin matrices will vanish.

Proposition 4 Let 1 ≤ k, l ≤ n with k 6 = l. Then E det Λ

= 0 if and only if E det Λ

= 0.

Proof: Assume first that k = 1 and l = 2. Suppose that E det Λ

= 0 and let us prove that E det Λ

= 0. Since det Λ

= 0 a.s., from (9) we obtain

D

F D

G = D

F D

G a.s. (25) for any i, j ≥ 1 (recall that D

F = DF ⊗

e

). That is,

DF D

G = DGD

F a.s., (26)

for any i ≥ 1. Let us apply the divergence operator δ (the adjoint of D) to both members of equation (26). From (8) we obtain δDF = nF and δDG = nG.

Using Proposition 1.3.3 in [5], we get

nF D

G − h DF, DD

G i

= nGD

F − h DG, DD

F i

a.s., which can be written as (using the notation (7))

I

(f )I

(g

) − (n − 1)

I

(g

)I

(f

)

= I

(g)I

(f

) − (n − 1)

I

(f

)I

(g

) a.s.

By the product formula (5), the above relation becomes I

(f ⊗ ˜ g

) +

n−1X

k=1

(k!C

C

− (n − 1)(k − 1)!C

C

)

× I

(f ⊗ ˜

g

)

= I

(g ⊗ ˜ f

) +

n−1X

k=1

(k!C

C

− (n − 1)(k − 1)!C

C

)

× I

(g ⊗ ˜

f

) a.s.

By identifying the terms in each Wiener chaos, we obtain

f ⊗ ˜

g

= g ⊗ ˜

f

for any i ≥ 1 and for any k = 0, . . . , n − 1. A further application of the product formula for multiple integrals yields

F DG = GDF a.s.

We differentiate the above relation in the Malliavin sense and we have F D

G + D

F D

G = GD

F + D

GD

F a.s.

for every i, j ≥ 1. By (25),

F D

G = GD

F a.s.

and this clearly implies that det Λ

= 0 a.s.

Suppose now that E det Λ

= 0. Then Λ

= 0 a.s. and from (9) we get D

F D

G = D

F D

G a.s.

for any i, j, p, q ≥ 1. This implies

DD

F D

G = DD

GD

F a.s. (27) for any i, p, q ≥ 1 Applying the divergence operator δ to equation (27) yields (n − 1)D

F D

G −h DD

F, DD

G i

= (n − 1)D

GD

F −h DD

G, DD

F i

a.s. This equality can be written as

I

(f

)I

(g

) − (n − 2)

I

(g

)I

(f

)

= I

(g

)I

(f

) − (n − 2)

I

(f

)I

(g

) a.s.

By the product formula for multiple integrals we get for every j, p, q ≥ 1 I

f

⊗ ˜ g

+

n−2X

k=1

h

k!C

C

− (n − 2)(k − 1)!C

C

× I

f

⊗ ˜

g

= I

g

⊗ ˜ f

+

n−2X

k=1

h

k!C

C

− (n − 2)(k − 1)!C

C

× I

g

⊗ ˜

f

a.s.

Identifying the coefficients of each Wiener chaos we obtain

f

⊗ ˜

g

= g

⊗ ˜

f