Wiener-It？ Chaos Expansion of Hilbert Space Valued Random Variables

The notion of -fold iterated It？ integral with respect to a cylindrical Hilbert space valued Wiener process is introduced and the Wiener-It？ 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-It？ 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-It？ chaos expansion for Hilbert space valued random variables. In order to achieve this, we first prove the Hilbert space valued version of the It？ representation theorem in Section 2. This generalization is established in Theorem 9 and Corollary 11. In Section 3, we introduce iterated It？ integrals and multiple It？ integrals with respect to a cylindrical Wiener process. In the Hilbert space valued case, the integrand of an -fold iterated It？ integral is a function defined on a certain subset of 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-It？ chaos expansion in the -valued case in [5]. 2. It？ Representation Theorem for Hilbert Space Valued Random Variables Let be a probability space. For any separable Hilbert