Stochastic Differential Equations: A Wiener Chaos Approach
Abstract
A new method is described for constructing a generalized solution for stochastic differential equations. The method is based on the Cameron-Martin version of the Wiener Chaos expansion and provides a unified framework for the study of ordinary and partial differential equations driven by finite- or infinite-dimensional noise with either adapted or anticipating input. Existence, uniqueness, regularity, and probabilistic representation of this Wiener Chaos solution is established for a large class of equations. A number of examples are presented to illustrate the general constructions. A detailed analysis is presented for the various forms of the passive scalar equation and for the first-order It\^{o} stochastic partial differential equation. Applications to nonlinear filtering if diffusion processes and to the stochastic Navier-Stokes equation are also discussed.
Keywords
Cite
@article{arxiv.math/0504559,
title = {Stochastic Differential Equations: A Wiener Chaos Approach},
author = {S. V. Lototsky and B. L. Rozovskii},
journal= {arXiv preprint arXiv:math/0504559},
year = {2007}
}