部分可观的带随机跳正倒向随机系统的最优控制理论及其应用

With respect to theorical innovation, this project aims to study optimal control problem of possibly degenerate forward-backward stochastic systems under partial information by employing the elementary methods including theories like stochastic control, stochastic filtering, stochastic analysis, convex analysis, spike variation, Ekeland variational principle. A linear quadratic optimal control problem is investigated, meanwhile, a state feedback form of an optimal control and the solvability of the Riccati equation are discussed. Under the situation when correlated noises may be present between the state and the observation，the state equation may be with random jumps and the draft coefficient of the observation equation may be linear growth, stochastic maximum principle and verification theorem for optimal control are emphasized especially. With the help of theories of stochastic flows and adjoint vector fields, the relations among adjoint processes are established. Conditional probability density function of backward stochastic filtering satisfies a backward stochastic partial differential equation and corresponding solvability is researched. Moreover, the observable characterization of maximum principle is given..With respect to practical applications, by exersising the theoretical conclusions obtained, this project will consider the optimization arising in finace market, including problems like partially observable optimal portfolio and consumption selection, mean-variance hedge, risk indifference pricing. Explicit expressions or numerical results of optimal solutions for these problems are tried to derive, which are hoped to provide investors and consumers with reasonable advices..This project possesses very important theoretical meaning and application value for the investigation of stochastic optimal control problems under partial information.

理论方面：本项目拟运用随机控制、随机滤波、随机分析、凸分析、针状变分、艾克兰变分原理等理论工具，研究部分可观测信息下可退化的正倒向随机最优控制问题，并在此基础上讨论其线性二次最优控制的状态反馈形式及相应的黎卡提方程的可解性。重点研究系统状态和观测含有相关噪声、状态方程带有随机跳跃、观测方程漂移项系数线性增长时最优控制的随机最大值原理和验证定理，借助随机流和对偶向量场等理论确立对偶过程之间的联系，研究倒向滤波的条件概率密度函数满足的倒向随机偏微分方程的可解性，给出最大值原理的可观测刻画。.应用方面：本项目拟运用所得理论结果研究一些源自金融市场的实际问题，比如部分可观测信息下最优投资组合与消费选择、均值-方差对冲、风险中性定价等优化问题，争取得到最优解，导出其显式表达或者数值解，为投资者和消费者提供合理化建议。.本项目拟研究的内容对于部分可观随机最优控制问题的研究具有重要的理论意义和应用价值。

正倒向随机系统在随机控制、微分对策、递归效用、数理金融、偏微分方程等领域有着广泛的实际应用，同时系统处于部分信息情形或状态部分可观测情形也非常接近真实的金融市场情况。基于此，本项目运用随机控制、随机滤波、随机分析、凸分析、泛函分析等理论工具，主要研究系统处于部分信息情形或状态部分可观测情形下，可退化的正倒向随机系统的控制理论，特别是随机系统的最优控制和微分对策理论，同时尝试将其理论结果应用到一些金融实际问题中去。到目前为止，项目组在Journal of Optimization Theory and Applications，Mathematical Problems in Engineering，ESAIM: Control, Optimisation and Calculus of Variations，Stochastics and Dynamics等国际杂志共发表了6篇学术论文，其中SCI收录5篇，EI收录1篇。本项目部分重要研究成果概述如下：研究了跳扩散系统当系统状态和观测含有相关噪声时的控制问题，得到了最优控制的最大值原理和验证定理；研究了部分信息下正倒向重随机系统的微分对策问题，导出了纳什均衡点满足的充分必要条件；研究了几类非对称信息情形下线性二次非零和随机微分对策问题，得到了纳什均衡点的充分必要条件，并给出了纳什均衡点的显式反馈表达形式；研究了有限时间情形以及部分信息下无穷时间情形完全耦合的正倒向随机系统的最优控制问题，得到了关于最优控制的两个充分性条件，并将其应用于解决现金管理、风险最小化两个实际问题。上述研究结果丰富了正倒向随机系统最优控制和微分对策理论，为解决一些金融实际问题提供了潜在的理论工具，并为今后借助随机流和对偶向量场等理论建立最大值原理中对偶变量间的联系奠定了研究基础。