正倒向系统相关的偏微分方程与随机控制问题

项目名称： 正倒向系统相关的偏微分方程与随机控制问题

项目编号： No.11201268

项目类型： 青年科学基金项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 张峰

作者单位： 山东财经大学

项目金额： 22万元

中文摘要： 本项目旨在以倒向随机微分方程理论为工具研究与正倒向系统相关的偏微分方程与随机最优控制问题。考虑几类带非Lipschitz系数的二阶拟线性偏微分方程以及拟线性偏微分方程的障碍问题，研究其Sobolev弱解的存在唯一性并给出概率解释。分别用动态规划原理和最大值原理研究带脉冲控制的正倒向系统的随机最优控制问题，并且探讨理论结果在金融优化等问题中的应用。本项目研究不仅有重要的理论意义，也有重要的应用价值。研究结果将不仅能推广非线性Feynman-Kac公式，丰富偏微分方程的弱解理论，还能丰富随机控制理论，并且对倒向随机微分方程理论的研究也是有意义的补充。

中文关键词： 倒向随机微分方程；随机最优控制；最大值原理；比较定理；偏微分方程

英文摘要： This research program concerns partial differential equations and stochastic optimal control problems of forward-backward systems based on the theory of backward stochastic diferential equations. We study second order quasilinear partial differential equations and obstacle problems for quasilinear partial differential equations with non-Lipschitz coefficients. Existence and uniqueness as well as probabilistic interpretation of the Sobolev weak solutions will be obtained. We also study stochastic optimal control problems of forward-backward systems involving impulse controls by maximum principle as well as dynamic programming principle, and applications in mathematical finance will be also investigated. This research program is of significant value both in theory and in practice. The results will not only generalize the celebrated nonlinear Feynman-Kac formula and enrich the theory of weak soution of partial differential equations, but also enrich the theory of stochastic optimal control. Moreover, they will also be meaningful supplementary for the theory of backward stochastic differential equations.

英文关键词： backward stochastic differential equation；stochastic optimal control；maximum principle；comparison theorem；partial differential equation