The notion of n-fold iterated Itô integral with respect to a cylindrical Hilbert space valued Wiener process is introduced and the Wiener-Itô chaos expansion is obtained for a square Bochner integrable Hilbert space valued random variable. The expansion can serve a basis for developing the Hilbert space valued analog of Malliavin calculus of variations which can then be applied to the study of stochastic differential equations in Hilbert spaces and their solutions.

1. Introduction

The Wiener-Itô chaos expansion of a square integrable random variable which was first proved in [1] plays fundamental role in Malliavin calculus of variations [2, 3] which appeared to be a powerful instrument in the analysis of functionals of Brownian motion. The Malliavin calculus has found extensive applications to stochastic differential equations arising as models of various random phenomena. One of the important sources of such equations is markets modeling in financial mathematics [4, 5].

In the last decades, many researchers’ interest has been drawn to stochastic differential equations in infinite dimensional Hilbert spaces driven by a cylindrical Wiener process or, equivalently, by countable set of Brownian motions [6, 7]. For example, in financial mathematics, such equations are used in modeling interest rates term structure or zero-coupon bond market [8, 9]. The present work was motivated by the need to make the Malliavin calculus applicable to Hilbert space valued stochastic processes. The first step in this direction is to obtain the generalization of the Wiener-Itô chaos expansion for Hilbert space valued random variables.

In order to achieve this, we first prove the Hilbert space valued version of the Itô representation theorem in Section 2. This generalization is established in Theorem 9 and Corollary 11.

In Section 3, we introduce iterated Itô integrals and multiple Itô integrals with respect to a cylindrical Wiener process. In the Hilbert space valued case, the integrand of an n-fold iterated Itô integral is a function defined on a certain subset of ℝn and taking values in a certain space of Hilbert space valued continuous n-linear forms defined on the nth Cartesian power of the Hilbert space where the Wiener process takes values.

Section 4 contains main results of the paper which are stated in Theorem 9 and Corollary 11. The proof of the theorem follows the scheme of the proof of the Wiener-Itô chaos expansion in the ℝ-valued case in [5].

2. Itô Representation Theorem for Hilbert Space Valued Random Variables

Let (Ω,ℱ,P) be a probability space. For any separable Hilbert space ℍ, we denote by L2(Ω,ℱ,P;ℍ), or L2(Ω;ℍ) for short, the space of all Bochner square integrable ℍ-valued random variables on (Ω,ℱ,P). Let {ej} be an orthonormal basis in ℍ and let {βj(t)∣t∈ℝ}j=1∞ be a sequence of independent identically distributed Brownian motions on the probability space. Consider the corresponding cylinder Wiener process, defined by
(1)W(t)=∑j=1∞βj(t)ej,t∈ℝ.
It is easy to see that the series is not convergent in L2(Ω;ℍ); however, for any x∈ℍ, we have that (W(t),x):=∑j=1∞βj(ej,x)ℍ is a random variable belonging to L2(Ω;ℝ). Denote by ℬta,t≥a the σ-algebra, generated by the Wiener process at [a;t], that is, the σ-algebra, generated by the set of random variables (W(s),x), where a≤s≤t, x∈ℍ. (Note that the series (1) is not convergent in L2(Ω;ℍ) at t≠0 although (W(s),x)∈L2(Ω;ℝ) for any s∈ℝ, x∈ℍ.) The family {ℬta} is called the filtration generated by the Wiener process W(t),t≥a. Note that the Brownian motions βj(t),j∈ℕ,t≥a are martingales with respect to the filtration ℬta.

Let H be another separable Hilbert space and let {gi}i=1∞ be an orthonormal basis in H. Then, the family of operators {gi⊗ej}i,j=1∞, defined by the equality
(2)(gi⊗ej)x:=gi(ej,x)ℍ,x∈ℍ,
forms an orthonormal basis in the space ℒ2(ℍ,H) of all Hilbert-Schmidt operators acting from ℍ to H. Any A∈ℒ2(ℍ,H) has the decomposition
(3)A=∑i,j=1∞aij(gi⊗ej),
where aij=(gi,Aej)H and
(4)∥A∥ℒ2(ℍ,H)2=∑i,j=1∞|aij|2<∞.
For any ℒ2(ℍ;H)-valued random process φ(t) adapted to the filtration ℬta, where a≤t≤b, satisfying the property
(5)E[∫ab∥φ(t)∥ℒ2(ℍ,H)2dt]<∞,
the stochastic Itô integral with respect to the cylindrical Wiener process
(6)∫abφ(t,ω)dW(t)
is well-defined and is an element of the space L2(Ω;H) (see the definition and the properties in [6]). Note that if the function φ(t)=φ(t,ω) is ℬ(ℝ)×ℱ-measurable, the equality (5) implies
(7)E[∫ab∥φ(t)∥ℒ2(ℍ,H)2dt]=∫abE[∥φ(t)∥ℒ2(ℍ,H)2]dt.

Theorem 1.

For any ℬba-measurable random variable F∈L2(Ω;H), there exists a unique ℬta-adapted ℒ2(ℍ,H)-valued random process φ(t) satisfying (5) such that
(8)F=E[F]+∫abφ(t,ω)dW(t),∥F∥L2(Ω;H)2=∥E[F]∥H2+E[∫ab∥φ(t)∥ℒ2(ℍ,H)2dt].

Proof.

For any i∈ℕ, we have Fi∶=(F,gi)H∈L2(Ω;ℝ). Therefore, we have the following decomposition for F:
(9)F=∑i=1Figi.
Denote Fin=E[Fi∣ℬba,(n)], where ℬba,(n) is the filtration generated by the n-dimensional Wiener process
(10)Wn(t)=∑j=1nβj(t)ej,a≤t≤b.
By the finite dimensional Itô representation theorem ([10], Theorem 4.3.3.), there exists a unique ℝn-valued random process
(11)φin(t)=[νi1(n)(t),…,νin(n)(t)],a≤t≤b,
such that the following conditions hold:

all the mappings νik(n):(t,ω)↦νik(n)(t,ω) are ℬ(ℝ)×ℱ-measurable,

E[∫abνik(n)(t)2dt]<∞,k=1,…,n,

the processes νik(n)(t),k=1,…,n are adapted to the filtration ℬta.

Moreover, we have
(12)Fin=E[Fin]+∫abφin(t)dBn(t)=E[Fi]+∑k=1n∫abνik(n)(t)dβk(t).
Here, Bn(t) denotes the n-dimensional Brownian motion
(13)Bn(t)=[β1(t)⋮βn(t)].
Since ℬba,(n)⊆ℬba,(m) for m>n, we have
(14)Fjn=E[Fj∣ℬba,(n)]=E[E[Fj∣ℬba,(m)]∣ℬba,(n)]=E[Fjm∣ℬba,(n)]=E[E[Fim]+∑k=1m∫abνik(m)(t)dβk(t)∣ℬba,(n)]=E[Fi]+∑k=1n∫abνik(m)(t)dβk(t).
It follows from the uniqueness of the representation (12) that νik(m)≡νik(n) for k≤n<m. Thus, the equality (12) can be rewritten as
(15)Fin=E[Fin]+∫abφin(t)dBn(t)=E[Fi]+∑k=1n∫abνik(t)dβk(t).
By the Jensen inequality, we have
(16)Fin2=E[Fi∣ℬba,(n)]2≤E[Fi2∣ℬba,(n)].
Therefore,
(17)E[Fin2]≤E[E[Fi2∣ℬba,(n)]]=E[Fi2]<∞.
Thus,
(18)E[Fin2]=E[(E[Fi]+∑k=1n∫abνik(t)dβk(t))2]=E[Fi]2+∑k=1nE[∫abνik2(t)dt]≤E[Fi2]<∞.
Consequently,
(19)∑k=1∞E[∫abνik2(t)dt]<∞.
Since ℬba,(n)⊆ℬba,(n+1) and ℬba=⋃n∈ℕℬba,(n), we have
(20)Fi=limn→∞E[Fi∣ℬba,(n)]=E[Fi]+∑k=1∞∫abνik(t)dβk(t),E[Fi2]=limn→∞E[Fin2]=E[Fi]2+∑k=1∞E[∫abνik2(t)dt].
For any N∈ℕ, it holds that
(21)E[∑i=1NFi2]=∑i=1NE[Fi]2+∑i=1N∑k=1∞E[∫abνik2(t)dt]≤E[∥F∥H2].
It follows from (20) that
(22)∑i=1∞Figi=∑i=1∞E[Fi]gi+∑i=1∞gi∑k=1∞∫abνik(t)dβk(t),
since the series in the right-hand side of the equality are convergent in L2(Ω;H) by the estimate (21). Thus, the equality (8) holds true. It also follows from this estimate by the Levy theorem that we can pass to the limits at N→∞ in the equality. As a result, we obtain
(23)E[∥F∥H2]=∑i=1∞E[Fi]2+∑i=1∞∑k=1∞E[∫abνik2(t)dt]=∥E[F]∥H2+E[∫ab∑i,k=1∞νik2(t)dt]=∥E[F]∥H2+E[∫ab∥φ(t)∥ℒ2(ℍ,H)2dt],
where
(24)φ(t)∶=∑i,k=1∞νik(t)(gi⊗ek).
By the polarization identity, we obtain the following assertion.

Corollary 2.

Let ξ(t) and η(t) be ℒ2(ℍ,H)-valued random processes, adapted to the filtration ℬta,a≤t≤b, such that
(25)∫abE[∥ξ(t)∥ℒ2(ℍ,H)2]<∞,∫abE[∥η(t)∥ℒ2(ℍ,H)2]<∞.
Then,
(26)E[(∫abξ(t)dW(t),∫abη(t)dW(t))H]=∫abE[(ξ(t),η(t))ℒ2(ℍ,H)]dt.

3. Multiple Itô Integrals with respect to the Cylindrical Wiener Process

Let ξ(t1,t2,ω) be a function defined for a≤t1≤t2≤b, ω∈Ω taking values in the space ℒ2(ℍ;ℒ2(ℍ;H)) and ℬ(ℝ2)×ℱ-measurable. If for any t2∈[a;b] the random process ξ(t,t2),a≤t≤t2 is adapted to the filtration ℬta,0≤t≤t2 and the condition
(27)E[∫at2∥ξ(t,t2)∥ℒ2(ℍ;ℒ2(ℍ;H))2dt]<∞
holds true, then the Itô integral
(28)∫at2ξ(t1,t2)dW(t1)
is well defined and is an element of the space L2(Ω;ℒ2(ℍ;H)) for all t2∈[a;b]. As a function of t2, it is an ℒ2(ℍ;H)-valued random process adapted to the filtration ℬta. We also have
(29)E[∥∫at2ξ(t1,t2)dW(t1)∥ℒ2(ℍ;H)2]=E[∫at2∥ξ(t1,t2)∥ℒ2(ℍ;ℒ2(ℍ;H))2dt1]<∞.
If moreover we have
(30)E[∫ab∫at2∥ξ(t1,t2)∥ℒ2(ℍ;ℒ2(ℍ;H))2dt1dt2]<∞,
then the following iterated Itô integral is well defined:
(31)∫ab∫at2ξ(t1,t2)dW(t1)dW(t2)
and it satisfies the equality
(32)E[∥∫ab∫at2ξ(t1,t2)dW(t1)dW(t2)∥H2]=E[∫ab∫at2∥ξ(t1,t2)∥ℒ2(ℍ;H)2dt1dt2].
Note that the operators gi⊗ej⊗ek defined by the equality
(33)(gi⊗ej⊗ek)x∶=(gi⊗ej)(ek,x)ℍ,i,j,k∈ℕ
form an orthonormal basis in the space ℒ2(ℍ;ℒ2(ℍ;H)) and any operator A∈ℒ2(ℍ;ℒ2(ℍ;H)) has the following decomposition:
(34)A=∑i,j,kaijk(gi⊗ej⊗ek),∥A∥ℒ2(ℍ;ℒ2(ℍ;H))2=∑i,j,kaijk2<∞.
We can identify the decomposition (34) with the H-valued bilinear form on ℍ×ℍ defined by the equality (gi⊗ej⊗ek)(x,y)∶=gi(ej,x)ℍ(ek,y)ℍ. Thus, the space ℒ2(ℍ;ℒ2(ℍ;H)) can be identified with the Hilbert space ℒ22(ℍ×ℍ;H) of all continuous bilinear forms A of the form (34) with the norm generated by the scalar product
(35)(A,B)ℒ22(ℍ×ℍ;H)2=∑i,j,kaijkbijk,
where A=∑i,j,kaijk(gi⊗ej⊗ek), B=∑i,j,kbijk(gi⊗ej⊗ek). So, the iterated integral (31) with respect to the cylindrical Wiener process W(t) is well defined for any ℒ22(ℍ×ℍ;H)-valued ℬ(ℝ2)×ℱ-measurable function ξ(t1,t2,ω), defined on S2a,b×Ω, where
(36)S2a,b={(t1,t2)∣a≤t1≤t2≤b},
if it satisfies the condition (30).

One can easily extend the above definition to the case of arbitrary n>2, defining the n times iterated integral
(37)Jn(ξ)=∫ab∫at2…∫atn-1ξ(t1,t2,…,tn,ω)dW(t1),…,dW(tn)
inductively for any function ξ:Sn×Ω→ℒ2n(ℍ×n;H), where
(38)Sna,b={(t1,…,tn)∣a≤t1≤,…,tn≤b},ℒ2n(ℍ×n;H) is the space of all continuous H-valued n-linear forms on ℍ×⋯×ℍ having form
(39)A=∑i,k1,…,knai,k1,…,kn(gi⊗ek1⊗,…,⊗ekn)
such that
(40)∥A∥ℒ2n(ℍ×n;H)2=∑i,k1,…,knai,k1,…,kn2<∞,
if it satisfies the following conditions:

ξ is ℬ(ℝn)×ℱ-measurable;

ξ(t1,t2,…,tn) is ℬt1a-measurable for any (t1,t2,…,tn)∈Sna,b;

For the defined iterated integral, we have
(41)E[∥Jn(ξ)∥H2]=∫ab∫at2⋯∫atn-1E[∥ξ(t1,…,tn)∥ℒ2n(ℍ×n;H)2]dt1,…,dtn.
We state without proof the next two lemmas which are straightforward generalizations of the corresponding properties of the ℝ-valued iterated Itô integrals.

Lemma 3.

Let ξ:Sn1a,b×Ω→ℒ2n1(ℍ×n1;H) and η:Sn2a,b×Ω→ℒ2n2(ℍ×n2;H) satisfy the conditions (n-i), (n-ii) and (n-iii) with n=n1 and n=n2 correspondingly. Then,
(42)E[(Jn1(ξ),Jn2(η))H]={0,n1≠n2,(ξ,η)L2(Sna,b;ℒ2n(ℍ×n;H)),n1=n2=n.

Lemma 4.

Let ξ:Sna,b×Ω→ℒ2n(ℍ×n;H) satisfy the conditions (n-i), (n-ii) and (n-iii). Then,
(43)E[Jn(ξ)|ℬta,b]=∫at∫at2⋯∫atn-1ξ(t1,t2,…,tn,ω)dW(t1),…,dW(tn).
Denote by L^2([a;b]n;ℒ2n(ℍ×n;H)) the space of all symmetric functions f∈:[a;b]n→ℒ2n(ℍ×n;H) satisfying the condition
(44)∥f∥L2([a;b]n;ℒ2n(ℍ×n;H))2=∫⋯[a;b]n∫∥fn(t1,…,tn)∥ℒ2n(ℍ×n;H)2dt1,…,dtn<∞.

Definition 5.

For any f∈L^2([a;b]n;ℒ2n(ℍ×n;H)), define the multiple n-fold Itô integral by the equality
(45)In(f)∶=n!Jn(f),
if the right-hand side iterated integral exists.

4. The Decomposition Theorem

To establish the main result, we need a few lemmas.

Lemma 6.

The set of random variables
(46){φ(βj1(ti1),…,βjn(tin))|jk∈ℕ,tk∈[a;b],ssφ∈C0∞(ℝn),n∈ℕtintin}
is dense in L2(ℬba,P).

Proof.

Let {ti}i=1∞ be a dense subset in [a;b]. Let {ζk}k=1∞ be a fixed ordering of the countable set of random variables {βj(ti)}j,i=1∞. Denote by ℋp be the σ-algebra generated by {ζk}k=1p. We have ℋp⊂ℋp+1 for all p∈ℕ and ℬba is the smallest σ-algebra containing all ℋp.

For any g∈L2(ℬba,P), we have
(47)g=E(g∣ℬba)=limp→∞E(g∣ℋp),
where the limit is pointwise a.e. with respect to P and in L2(ℬba,P). By the Doob-Dynkin lemma for any p∈ℕ, there exists a Borel function gp:ℝp→ℝ such that
(48)E[g∣ℋp]=gp(ζ1,…,ζp).
Let Pζ1,…,ζp be the probability measure on ℝp generated by ζ1,…,ζp. Since gp can be approximated in L2(ℝp,Pζ1,…,ζp) by functions φ∈C0∞(ℝp), the assertion follows.

Consider the following exponential functionals of Brownian motions βj:
(49)ℰθj,a,b∶=exp[∫abθ(t)dβj(t)-|θ|022],sssssssssssssj∈ℕ,θ∈L2[a;b].

Lemma 7.

The linear span of the set
(50){∏k=1pℰθkjk,a,b|jk∈ℕ,θk∈L2[a;b],k=1,…,p,p∈ℕ}
is dense in L2(ℬba,P).

Proof.

Let g∈L2(ℬba,P) be orthogonal to all functions of the family (50). Take θk(t)=λk1[a;tk], where {tk}k=1p⊂[a;b], λ¯=(λ1,…,λp)∈ℝp. Then, we have
(51)∏k=1pℰθkjk,a,b=∏k=1pexp[λk∫atkdβjk(t)-tk-a2]=exp[-12∑k=1p(tk-a)]·exp[∑k=1pλkβjk(tk)].
It follows that
(52)G(λ¯)∶=∫Ωexp[∑k=1pλkβjk(tk)]g(ω)dP(ω)=0
for all p∈ℕ, λ¯∈ℝp, j1,…,jp∈ℕ, t1,…,tp∈[a;b].

Fix p,j1,…,jp,t1,…,tp. The corresponding function G(λ¯) is real analytic. Consider its analytic extension onto ℂp. It is also equal to zero. Taking λ¯=(iy1,…,iyp), we obtain
(53)∫Ωexp[i∑k=1pykβjk(tk)]g(ω)dP(ω)=0.
For any φ∈C0∞(ℝp) denoting by φ^ its Fourier transform, we obtain
(54)∫Ωφ(βj1(t1),…,βjp(tp))g(ω)dP(ω)=∫Ω1(2π)p/2ssssssss×∫ℝpφ^(y¯)exp[i∑k=1pykβjk(tk)]dy¯g(ω)dP(ω)=1(2π)p/2∫ℝpφ^(y¯)×∫Ωexp[i∑k=1pykβjk(tk)]g(ω)dP(ω)dy¯=0.
Thus, g is orthogonal to a dense subset in L2(ℬba,P).

Lemma 8.

For any α1,…,αn∈ℕ∪{0}, θ1,…,θn∈𝒮, the product
(55)∏j=1nhαj(∫abθj(t)dβj(t)|θj·1[a;b]|0)
is a linear combination of the iterated Itô integrals of the form
(56)∫ab∫atp⋯∫at2θj1(t1),…,θjp(tp)dβj1(t1),…,dβjp(tp),
where p=α1+⋯+αn and j1,…,jp∈{1,…,n}.

Proof.

We use induction with respect to p. The assertion is evident for p=1. Suppose it is true for some p∈ℕ. Let α1+⋯+αn=p+1. We introduce the following notation:
(57)Xj(t)∶=|θj·1[a;t]|0αjαj!hαj(∫atθj(τ)dβj(τ)|θj·1[a;t]|0)=∫at∫atαj⋯∫at2θj(t1),…,θj(tαj)dβj(t1),…,dβj(tαj)=:∫atYj(tαj,ω)dβj(tαj),j=1,…,n.
This means that Xj satisfy the following stochastic differential equations:
(58)dXj(t)=Yj(t,ω)dβj(tαj),j=1,…,n,
and we have
(59)Yj(t)=|θj·1[a;t]|0αj-1(αj-1)!hαj-1(∫atθj(τ)dβj(τ)|θj·1[a;t]|0).
By the Itô formula, we obtain
(60)d(∏j=1nXj(t))=∑j=1n[∏i≠jXi(t)]dXj(t)+∑i≠jdXi(t)·dXj(t).
Since dXi(t)·dXj(t)=Yi(t,ω)Yj(t,ω)dβi(t)dβj(t)=0, we come to the equality
(61)∏j=1nXj(b)=∑j=1n∫ab[∏i≠jXi(t)]Yj(t,ω)dβj(t)=∑j=1n∫ab[∏i≠j|θj·1[a;t]|0αjαjhαj(∫atθj(τ)dβj(τ))ssssssssss·|θj·1[a;t]|0αj-1(αj-1)!hαj-1(∫atθj(τ)dβj(τ)|θj·1[a;t]|0)]×θj(t)dβj(t).
Since ∑i≠jαi+αj-1=p, we can apply the assumption to the products in the brackets, thus completing the proof.

Theorem 9.

Any ℬba-measurable random variable F∈L2(Ω;H) has the unique decomposition
(62)F=∑n=0∞In(fn),
where fn∈L^2([a;b]n;ℒ2n(ℍ×n;H)),n∈ℕ, I0(f0):=f0∈H. It holds that
(63)∥F∥L2(Ω;H)2=∑n=0∞n!∥fn∥L2([a;b]n;ℒ2n(ℍ×n;H))2,
where ∥f0∥L2([a;b]0;ℒ20(ℍ×0;H))2∶=∥f0∥H2.

Proof.

By Theorem 1, we have
(64)F=E[F]+∫abφ1(t1,ω)dW(t1),
where φ1(t) is a ℒ21(ℍ;H)-valued random variable, ℬta-measurable for any t. For its norm, we have
(65)∥F∥L2(Ω;H)2=∥E[F]∥H2+E[∫ab∥φ1(t)∥ℒ2(ℍ;H)2dt1].
Let ψ0∶=E[F]. For any t1∈[a,b] by Theorem 1, taking t1 for b and ℒ2(ℍ;H) for H, we obtain the following representation:
(66)φ1(t1)=E[φ1(t1)]+∫at1φ2(t1,t2,ω)dW(t2),
where φ2:S2a,b×Ω→ℒ2(ℍ;ℒ2(ℍ;H))≅ℒ22(ℍ×2;H), with the following equality for the norm:
(67)∥φ1(t1)∥ℒ2(Ω;ℒ2(ℍ;H))2=∥E[φ1(t1)]∥ℒ2(ℍ;H)2+E[∫at1∥φ2(t1,t2)∥ℒ22(ℍ×2;H)2dt2].
Set ψ1(t1)∶=E[φ1(t1)]. Substituting the representation (66) into (64), we come to the equality
(68)F=J0(ψ0)+∫abψ1(t1)dW(t1)+∫ab∫at1φ2(t1,t2,ω)dW(t2)dW(t1).
From (65) and (67), it follows that
(69)∥F∥L2(Ω;H)2=∥ψ0∥H2+∫ab∥ψ1(t1)∥ℒ21(ℍ;H)2dt1+E[∫ab∫at1∥φ2(t1,t2)∥ℒ22(ℍ×2;H)2dt2dt1]=∥ψ0∥H2+∫ab∥ψ1(t1)∥ℒ21(ℍ;H)2dt1+∫ab∫at1E[∥φ2(t1,t2)∥ℒ22(ℍ×2;H)2]dt2dt1.
Applying further in similar manner Theorem 1 and setting
(70)ψn(t1,…,tn)∶=E[φn(t1,…,tn)]∈ℒ2n(ℍ×n;H),hhhhhhhhhhhhhhhhhhhhhhhh(t1,…,tn)∈Sna,b
at the nth step of this process, we obtain
(71)F=J0(ψ0)+J1(ψ1)+⋯+Jn(ψn)+Jn+1(φn+1)
with
(72)∥F∥L2(Ω;H)2=∥ψ0∥H2+∫ab∥ψ1(t1)∥ℒ21(ℍ;H)2dt1+∫ab∫at1∥ψ2(t1,t2)∥ℒ22(ℍ×2;H)2dt2dt1+⋯+∫ab∫at1…∫atn-1∥ψn(t1,…,tn)∥ℒ2n(ℍ×n;H)2dtn,…,dt1+∫ab∫at1…∫atnE[∥φn+1(t1,…,tn+1)∥ℒ2n+1(ℍ×(n+1);H)2]dtn+1,…,dt1.
It follows from here that
(73)∥ψ0∥H2+∑k=1n∫ab∫at1⋯∫atk-1∥ψk(t1,…,tk)∥ℒ2k(ℍ×k;H)2dtk,…,dt1≤∥F∥L2(Ω;H)2
and consequently
(74)∥ψ0∥H2+∑k=1∞∫ab∫at1⋯∫atk-1∥ψk(t1,…,tk)∥ℒ2k(ℍ×k;H)2dtk,…,dt1<∞.
This means that the series
(75)∑n=0∞Jn(ψn)
is convergent in the space L2(Ω;H) and
(76)Φ∶=limn→∞Jn+1(φn+1)
is an element of this space. By Lemma 3, the integral Jn+1(φn+1) is orthogonal in L2(Ω;H) to the integrals J0(ψ0),…,Jn(ψn) for all ψ0∈H, ψ1∈L2(S1a,b;ℒ21(ℍ;H)),…,ψn∈L2(Sna,b;ℒ2n(ℍ×n;H)). It follows that Φ is orthogonal to Jn(ψn) for all n∈ℕ∪{0}, ψn∈L2(Sna,b;ℒ2n(ℍ×n;H)).

Note that by the well-known connection between iterated Itô integrals with respect to a Brownian motion and the Hermite polynomials, we have
(77)Jn(gi⊗ej⊗nθ⊗n)=gi∫ab∫at1⋯∫atn-1θ⊗n(t1,…,tn)dβj(tn)…dβj(t1)=gi|θ·1[a;b]|0nhn(∫abθ(t)dβj(t)|θ·1[a;b]|0)
for any n,i,j∈ℕ, θ∈𝒮.

Let Φ=∑i=1∞Φigi, where Φi∈L2(Ω;ℝ). For any n,i,j1,…,jn∈ℕ and any θj1,…,θjn∈𝒮, we have
(78)E[Φi·∫ab∫atn⋯∫at2θj1(t1),…,θjn(tn)dβj1(t1),…,dβjn(tn)∫ab]=E[(Φ,gi∫ab∫atn⋯∫at2θj1(t1),…,θjn(tn)dβj1(t1),…,dβjn(tn)∫ab∫atn)H]=(Φ,Jn(gi⊗ej1⊗⋯⊗ejn⊗θj1⊗⋯⊗θjn))=0,
from where, by Lemma 8 and the equality (77), it follows that, for any i∈ℕ, Φ