Mathematical Sciences: Applications of Stochastic Calculus of Variations and Skorohod's Integral to Stochastic Systems

点击翻译按钮获取中文摘要

The investigator intends to do research on two topics using the methods of the stochastic calculus of variations. The first topic concerns the analysis of "acausal" stochastic differential equations driven by Brownian motion. Recently, Nualart and Pardoux used the stochastic calculus of variations to develop a calculus for the Skorokhod integral, which is an integral with respect to Brownian motion for integrands not necessarily adapted to the past of the Brownian motion. This should allow one to study equations containing "acausal" features, that is exhibiting dependence not only on the past but also on the future. Professor Ocone intends to undertake research in this direction. The second topic concerns the analysis of the probability distributions of solutions of stochastic partial differential equations. This study has implications for nonlinear filtering theory. In this project the investigator wishes to continue the line of investigation that leads to a study of regularity of densities for linear functionals on solutions to stochastic partial differential equations. He will also study the nonlinear filtering problem, and attempt to relate Wiener space gradients of conditional distributions of nonlinear filters to the issue of finite dimensional computability of filters. This is a theoretical work where computability is characterized by the existence of certain finite dimensional stochastic differential equation that governs certain conditional expectations. There is a possibility that the investigator can relate computability to certain rank conditions for Frechet derivatives of certain maps. This research belongs to a larger research effort on stochastic differential equations and filtering problems that is now an active area in applied probability.