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

研究者打算使用 随机变分法第一主题 涉及"因果"随机微分方程的分析 由布朗运动驱动。最近，Nualart和Pardoux使用了 随机变分法，以开发一种微积分， Skorokhod积分，这是关于布朗的积分 对于不一定适应过去的被整合者的动议 布朗运动这应该允许一个研究方程包含 “因果”特征，即表现出不仅依赖于 过去，也是未来。奥康教授打算 研究这个方向。第二个主题涉及分析 随机偏微分方程解的概率分布 微分方程这项研究对非线性 过滤理论在本项目中，研究者希望 继续进行调查， 解上线性泛函密度的正则性 随机偏微分方程他还将研究 非线性滤波问题，并试图将维纳空间 非线性滤波器的条件分布的梯度 滤波器的有限维可计算性问题。这是一 可计算性的特点是 一类有限维随机微分的存在性 控制某些条件期望的等式。有一个 研究者可以将可计算性与 某些映射的Frechet导数的秩条件。 这项研究属于一个更大的研究工作，随机 微分方程和过滤问题，现在是一个活跃的 在应用概率。

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.