Stochastic Equations in Formal Mappings

Abstract

Let us consider a nonlinear stochastic equation

$$\begin{array}{*{20}{c}} {y(t) = y(0) + \int\limits_{0}^{t} {a(\tau )(y(\tau ))d\tau + \int\limits_{0}^{t} {b(\tau )(y(\tau ))dw(\tau ),} } } & {0 \leqslant t \leqslant T} \\ \end{array}$$

(1.1)

in Hilbert space Y, where:

• w is a Wiener process, associated with canonical triple H ₊ ⊂ H ₀ ⊂ H_ with a Hilbert-Schmidt embedding (all Hilbert spaces are supposed to be real and separable);

• y is a non-anticipating process in Hilbert space Y;

• a and b are continuous mappings from [0, T] × Y into Y and ℒ₂ (Y) respectively.

This work was supported, in part, by the International Soros Science Education Program (ISSEP) through grant N PSU051117.