Stochastic Differential Equations: A Wiener Chaos Approach

By Sergey V. Lototsky, Boris L. Rozovskii. 2005

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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.

Subfields: Finance, Modeling and Simulation, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/a47f5558b28e1dda514e34a1bd84d05a73b7f6cb

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Citations in the corpus

Sparse tensor discretizations of high-dimensional parametric and stochastic PDEs* (2011, background): “…eye towards uncorrelated, or white noise inputs (see, e.g., Holden, Oksendal, Uboe and Zhang (1996), Da Prato and Zabczyk (1992), Da Prato (2006), Lototsky and Rozovskii (2006), Prévôt and Röckner (2007), Dalang, Khoshnevisan, Mueller, Nualart and Xiao (2009) and the references therein), PDEs…”