Stochastic integration of functions with values in a Banach space
Volume 166 / 2005
Abstract
Let $H$ be a separable real Hilbert space and let $E$ be a real Banach space. In this paper we construct a stochastic integral for certain operator-valued functions ${\mit\Phi}:(0,T)\to{\scr L}(H,E)$ with respect to a cylindrical Wiener process $\{W_H(t)\}_{t\in[0,T]}$. The construction of the integral is given by a series expansion in terms of the stochastic integrals for certain $E$-valued functions. As a substitute for the Itô isometry we show that the square expectation of the integral equals the radonifying norm of an operator which is canonically associated with the integrand. We obtain characterizations for the class of stochastically integrable functions and prove various convergence theorems. The results are applied to the study of linear evolution equations with additive cylindrical noise in general Banach spaces. An example is presented of a linear evolution equation driven by a one-dimensional Brownian motion which has no weak solution.
