倒向随机微分方程及其在最优控制中的应用
发布时间：2018-10-21 17:40
【摘要】：在实际生活中往往会遇到这样的问题:为达到将来某预定目标,如何确定当前的状态和实施战略,而倒向随机微分方程恰好是解决这类问题十分重要的方法。同时,倒向随机微分方程也是研究偏微分方程、随机控制、计算机科学等领域的有效工具。倒向随机微分方程现在已成为随机分析范畴非常重要的分支之一。本文是一篇综述报告,主要研究一类特殊的倒向随机微分方程(BSDE)在局部Lipschitz条件下解的存在性及唯一性,以及与之相关的最优控制随机系统理论——最大值原理。具体内容如下:第一章主要介绍倒向随机微分方程的发展状况、研究现状及理论意义;第二章为本文涉及到的相关预备知识;第三章研究一特殊类倒向随机微分方程在局部Lipschitz条件下局部解和全局解的存在性及唯一性;第四章着重探讨一类广义的带有随机系数的Riccati方程解的全局存在性;第五章给出正向和倒向状态方程最优控制系统的最大值原理;第六章回顾本论文的几个主要结论。
[Abstract]:In real life, we often encounter such problems: how to determine the current state and implementation strategy in order to achieve a predetermined goal in the future, and the backward stochastic differential equation is exactly the most important method to solve this kind of problems. At the same time, backward stochastic differential equation is also an effective tool to study partial differential equation, stochastic control, computer science and so on. The backward stochastic differential equation has become one of the most important branches in the category of stochastic analysis. In this paper, we study the existence and uniqueness of the solution of a special backward stochastic differential equation (BSDE) under the local Lipschitz condition, and the related optimal control stochastic system theory, the maximum principle. The main contents are as follows: the first chapter mainly introduces the development of backward stochastic differential equations, the research status and theoretical significance, the second chapter is the related preparatory knowledge involved in this paper; In chapter 3, the existence and uniqueness of local solution and global solution of a special class of backward stochastic differential equations under local Lipschitz conditions are studied, and the global existence of solutions of a class of generalized Riccati equations with stochastic coefficients is discussed in chapter 4. In chapter 5, the maximum principle of forward and backward state equation optimal control systems is given, and the main conclusions of this paper are reviewed in chapter 6.
【学位授予单位】：中国科学技术大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O211.63;O232
本文编号：2285864
[Abstract]:In real life, we often encounter such problems: how to determine the current state and implementation strategy in order to achieve a predetermined goal in the future, and the backward stochastic differential equation is exactly the most important method to solve this kind of problems. At the same time, backward stochastic differential equation is also an effective tool to study partial differential equation, stochastic control, computer science and so on. The backward stochastic differential equation has become one of the most important branches in the category of stochastic analysis. In this paper, we study the existence and uniqueness of the solution of a special backward stochastic differential equation (BSDE) under the local Lipschitz condition, and the related optimal control stochastic system theory, the maximum principle. The main contents are as follows: the first chapter mainly introduces the development of backward stochastic differential equations, the research status and theoretical significance, the second chapter is the related preparatory knowledge involved in this paper; In chapter 3, the existence and uniqueness of local solution and global solution of a special class of backward stochastic differential equations under local Lipschitz conditions are studied, and the global existence of solutions of a class of generalized Riccati equations with stochastic coefficients is discussed in chapter 4. In chapter 5, the maximum principle of forward and backward state equation optimal control systems is given, and the main conclusions of this paper are reviewed in chapter 6.
【学位授予单位】：中国科学技术大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O211.63;O232
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1 李菁;倒向随机微分方程理论及其在金融和行为金融中的应用[D];吉林大学;2016年
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