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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Stochastic integral representation of multiplicative operator functionals of a Wiener process


Author: Mark A. Pinsky
Journal: Trans. Amer. Math. Soc. 167 (1972), 89-104
MSC: Primary 60J60
MathSciNet review: 0295433
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Abstract: Let M be a multiplicative operator functional of (X, L) where X is a d-dimensional Wiener process and L is a separable Hilbert space. Sufficient conditions are given in order that M be equivalent to a solution of the linear Itô equation

$\displaystyle M(t) = I + \sum\limits_{j = 1}^d {\int_0^t {M(s){B_j}} } (x(s))d{x_j}(s) + \int_0^t {M(s){B_0}(x(s))ds,} $

where $ {B_0}, \ldots ,{B_d}$ are bounded operator functions on $ {R^d}$. The conditions require that the equation $ T(t)f = E[M(t)f(x(t))]$ define a semigroup on $ {L^2}({R^d})$ whose infinitesimal generator has a domain which contains all linear functions of the coordinates $ ({x_1}, \ldots ,{x_d})$. The proof of this result depends on an a priori representation of the semigroup $ T(t)$ in terms of the Wiener semigroup and a first order matrix operator. A second result characterizes solutions of the above Itô equation with $ {B_0} = 0$. A sufficient condition that M belong to this class is that $ E[M(t)]$ be the identity operator on L and that $ M(t)$ be invertible for each $ t > 0$. The proof of this result uses the martingale stochastic integral of H. Kunita and S. Watanabe.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1972-0295433-8
PII: S 0002-9947(1972)0295433-8
Keywords: Multiplicative operator functional, martingale stochastic integral, Hilbert-Schmidt operators, parabolic systems of equations
Article copyright: © Copyright 1972 American Mathematical Society