随机偏微分方程的随机表示理论及其应用

项目名称： 随机偏微分方程的随机表示理论及其应用

项目编号： No.11471079

项目类型： 面上项目

立项/批准年度： 2015

项目学科： 数理科学和化学

项目作者： 张奇

作者单位： 复旦大学

项目金额： 65万元

中文摘要： 本项目在前期工作的基础上，研究随机偏微分方程的解的表示理论及其应用。利用Markov系数的正倒向重随机微分方程和非Markov系数的正倒向随机微分积分方程，从两个途径开展研究，给出对应的随机偏微分方程的随机表示，分别称为Markov系数的随机表示定理和非Markov系数的随机表示定理。对于前者，将以随机分形方程为例证明它的随机表示定理，此类随机偏微分方程对应的正倒向重随机微分方程的可解性无法由鞅表示定理直接给出。对于后者，将证明倒向随机偏微分积分方程的随机表示定理，在此过程中，将首先解决可退化的倒向随机偏微分积分方程的可解性和正则性问题。随机偏微分方程的随机表示不仅是非线性Feynman-Kac公式理论在随机偏微分方程中的自然扩展，还可以应用于随机动力系统、金融数学等领域，本项目还将基于以上研究结果，将随机表示理论应用到非线性随机分形方程的平稳解、期权定价等问题中。

中文关键词： 随机偏微分方程；随机表示；随机分形方程；倒向随机偏微分（积分）方程；正倒向（重）随机微分（积分）方程

英文摘要： Based on the previous works, this project aims to study the theories and applications of stochastic representation for stochastic partial differential equation. We will carry out this project by two different methods, i.e. the forward-backward doubly stochastic differential equations with Markovian coefficients and the forward-backward stochastic integro-differential equations with non-Markovian coefficients. We prove the stochastic representations for the corresponding stochastic partial differential equations which are called the stochastic representation theory with Markovian coefficients and the stochastic representation theory with non-Markovian coefficients, respectively. For the former issue, we take the stochastic fractal equation for example to study the stochastic representation theory. The stochastic fractal equation belongs to a class of stochastic partial differential equations, for which the solvability of the corresponding forward-backward doubly stochastic differential equations cannot be derived directly due to the absence of the martingale representation theory. For the latter issue, we prove the stochastic representation theory of the backward stochastic partial integro-differential equation. For this, the solvability and regularity of the degenerate backward stochastic partial integro-differential equation will be studied first. The study of the stochastic representation for stochastic partial differential equation is not only a natural extension of non-linear Feynman-Kac formula to the stochastic partial differential equation, but also can be applied to some related research fields, such as the random dynamical system, mathematical finance, etc. In this project, we will also demonstrate the applications of our theoretical results of stochastic representation to the stationary solution of non-linear stochastic fractal equation and the option pricing theory.

英文关键词： stochastic partial differential equation;stochastic representation;stochastic fractal equation;backward stochastic partial (integro)-differential equation;forward-backward (doubly) stochastic (integro)-differential equation