Stochastic Differential Equations: A Wiener Chaos Approach

Details Stochastic Differential Equations: A Wiener Chaos Approach Stochastic Differential Equations: A Wiener Chaos Approach

2005-04-27

arxiv.org/abs/math/0504559v1

Mathematics

Probability

Scientific paper

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.

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