2005/04/27 by S. V. Lototsky, Lototsky, S. V., B. L. Rozovskii +1 · 1 citation
Mathematics · #60H15 #FOS: Mathematics #Probability (math.PR) #math.PR #msc:60H15
paper · pdf · doi:10.48550/arxiv.math/0504559
arxiv created 2005/04/27 · arxiv updated 2009/12/01
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 Itô stochastic partial differential equation. Applications to nonlinear filtering if diffusion processes and to the stochastic Navier-Stokes equation are also discussed.