Wiener-Itô Chaos Expansion of Hilbert Space Valued Random Variables

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.


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 zerocoupon 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 -fold iterated Itô integral is a function defined on a certain subset of R  and taking values in a certain space of Hilbert space valued continuous -linear forms defined on the th 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].

Itô Representation Theorem for Hilbert Space Valued Random Variables
Let (Ω, F, ) be a probability space.For any separable Hilbert space H, we denote by  2 (Ω, F, ; H), or  2 (Ω; H) for short, the space of all Bochner square integrable H-valued random variables on (Ω, F, ).Let {  } be an orthonormal basis in H and let {  () |  ∈ R} ∞ =1 be a sequence of independent identically distributed Brownian motions on the probability space.Consider the corresponding cylinder Wiener process, defined by It is easy to see that the series is not convergent in  2 (Ω; H); however, for any  ∈ H, we have that ((), ) := ∑ ∞ =1   (  , ) H is a random variable belonging to  2 (Ω; R).Denote by B   ,  ≥  the -algebra, generated by the Wiener process at [; ], that is, the -algebra, generated by the set of random variables ((), ), where  ≤  ≤ ,  ∈ H. (Note that the series (1) is not convergent in  2 (Ω; H) at  ̸ = 0 although ((), ) ∈  2 (Ω; R) for any  ∈ R,  ∈ H.) The family {B   } is called the filtration generated by the Wiener process (),  ≥ .Note that the Brownian motions   (),  ∈ N,  ≥  are martingales with respect to the filtration B   .Let  be another separable Hilbert space and let {  } ∞ =1 be an orthonormal basis in .Then, the family of operators , defined by the equality forms an orthonormal basis in the space L 2 (H, ) of all Hilbert-Schmidt operators acting from H to .Any  ∈ L 2 (H, ) has the decomposition where   = (  ,   )  and For any L 2 (H; )-valued random process () adapted to the filtration B   , where  ≤  ≤ , satisfying the property the stochastic Itô integral with respect to the cylindrical Wiener process is well-defined and is an element of the space  2 (Ω; ) (see the definition and the properties in [6]).Note that if the function () = (, ) is B(R)×F-measurable, the equality (5) implies Theorem 1.For any B   -measurable random variable  ∈  2 (Ω; ), there exists a unique B   -adapted L 2 (H, )-valued random process () satisfying (5) Proof.For any  ∈ N, we have   := (,   )  ∈  2 (Ω; R).Therefore, we have the following decomposition for : Denote , where B ,()  is the filtration generated by the -dimensional Wiener process By the finite dimensional Itô representation theorem ([10], Theorem 4.3.3.), there exists a unique R  -valued random process such that the following conditions hold: (i) all the mappings ] ()  : (, )  → ] ()  (, ) are B(R) × F-measurable, () 2 ] < ∞,  = 1, . . ., , (iii) the processes ] ()  (),  = 1, . . .,  are adapted to the filtration B   .Moreover, we have Here,   () denotes the -dimensional Brownian motion Since B ,()  ⊆ B ,()

𝑏
for  > , we have It follows from the uniqueness of the representation (12) that ] () ≡ ] ()  for  ≤  < .Thus, the equality (12) can be rewritten as By the Jensen inequality, we have Therefore, Thus, Consequently, Since B ,()  ⊆ B ,(+1)  and B   = ⋃ ∈N B ,()  , we have For any  ∈ N, it holds that It follows from (20) that since the series in the right-hand side of the equality are convergent in  2 (Ω; ) 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  → ∞ in the equality.As a result, we obtain where By the polarization identity, we obtain the following assertion.

Multiple Itô Integrals with respect to the Cylindrical Wiener Process
Let ( is well defined and is an element of the space  2 (Ω; L 2 (H; )) for all  2 ∈ [; ].As a function of  2 , it is an L 2 (H; )-valued random process adapted to the filtration B   .We also have If moreover we have then the following iterated Itô integral is well defined: if it satisfies the condition (30).One can easily extend the above definition to the case of arbitrary  > 2, defining the  times iterated integral if it satisfies the following conditions: For the defined iterated integral, we have We state without proof the next two lemmas which are straightforward generalizations of the corresponding properties of the R-valued iterated Itô integrals.

The Decomposition Theorem
To establish the main result, we need a few lemmas.
For any  ∈  ∞ 0 (R  ) denoting by φ its Fourier transform, we obtain Thus,  is orthogonal to a dense subset in This means that   satisfy the following stochastic differential equations: and we have By the Itô formula, we obtain () ⋅   () . (60) Since   () ⋅   () =   (, )  (, )  ()  () = 0, we come to the equality Since ∑  ̸ =    +   − 1 = , we can apply the assumption to the products in the brackets, thus completing the proof.