Abstract
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.
Highlights
The Wiener-Itochaos 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]
Summary
The Wiener-Itochaos 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. This generalization is established in Theorem 9 and Corollary 11. The proof of the theorem follows the scheme of the proof of the Wiener-Itochaos expansion in the R-valued case in [5]
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