Stochastic Differential Equations: A Wiener Chaos Approach [chapter]

Sergey Lototsky, Boris Rozovskii
2006 From Stochastic Calculus to Mathematical Finance  
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
more » ... d 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.
doi:10.1007/978-3-540-30788-4_23 fatcat:cnfbhbk52nblhevla7m3q57ore