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

研究者打算用随机变分法对两个课题进行研究。第一个主题涉及布朗运动驱动的“因果”随机微分方程的分析。最近，Nualart和Pardoux使用随机变分法发展了Skorokhod积分法，这是一个关于布朗运动的积分，不一定适应于布朗运动的过去。这应该允许人们研究包含“因果”特征的方程，这不仅表现出对过去的依赖，也表现出对未来的依赖。Ocone教授打算在这个方向上进行研究。第二个主题是分析随机偏微分方程解的概率分布。本研究对非线性滤波理论有一定的启示。在这个项目中，研究者希望继续研究导致随机偏微分方程解的线性泛函密度规律性的研究。他还将研究非线性滤波问题，并尝试将非线性滤波器的条件分布的维纳空间梯度与滤波器的有限维可计算性问题联系起来。这是一项理论工作，其中可计算性的特点是存在某些有限维随机微分方程，该方程控制某些条件期望。研究者有可能将某些映射的Frechet导数的可计算性与某些秩条件联系起来。这项研究属于随机微分方程和滤波问题的一个更大的研究工作，这是目前应用概率论的一个活跃领域。

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.