逐段决定马氏过程的测度值生成元与可加泛函

项目名称： 逐段决定马氏过程的测度值生成元与可加泛函

项目编号： No.11471218

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： 刘国欣

作者单位： 石家庄铁道大学

项目金额： 78万元

中文摘要： 逐段决定马氏过程，简记为PDMPs，是一类广泛的非扩散马氏过程。其随机性只限于随机跳时与随机跳转，在两个随机跳之间的轨道沿半动力系统演化。本项目着力于超出Davis的PDMPs范围的一般PDMPs的理论研究。从其基本特征--半动力系统特征出发，研究半动力系统的所谓可加泛函的相关性质。借助此可加泛函，给出PDMPs的可加泛函的表示；应用半动力系统可加泛函与沿动力系统轨道符号测度的一一对应关系，引入测度值算子的概念；再将PDMPs广生成元的概念推广为测度值生成元，使其定义域缩小到一定程度可退化为广生成元。进而建立一般PDMPs的测度值生成元的一般理论，包括Ito型公式等。 由于随机控制理论对Ito公式的本质依赖关系，把测度值生成元理论应用于PDMPs控制理论的研究，将HJB方程、QVI HJB方程统一为测度值HJB方程，建立一般PDMPs的HJB的统一理论框架。

中文关键词： 马氏过程；随机分析；生成元；可加泛函；随机最优控制

英文摘要： Piecewise deterministic Markov processes, PDMPs for short, are a general class of non-diffusion Markov processes, for which their randomness is only on the jump times and the post-jump locations. A PDMP evolves as a semi-dynamic system between jumps. This project aims at general PDMPs theory which is far beyond of Davis' PDMPs. Beginning with a semi-dynamic system,the one in the characteristic triple of a PDMP, we study the properties of the so-called additive functional of a semi-dynamic system. The additive functional of a PDMP can be represented by a additive functional of the semi-dynamic system. Based on the one-to-one relationship between the additive functional and the sign measure along the trajectories of a semi-dynamic system, we introduce a measure-valued operator. We will extend the concept of the extened generator into the one of the measure-valued generator of a PDMP such that the measure-valued generator can degenerate to extended generator by limitting its domain. Furthermore we'll establish the theory of measure-valued generators for general PDMPs, such as Ito-type formula in the general case. As we know, the stochastic optimal control is intrinctly on the Ito-type formula. By virtue of the theory of the measure-valued generators, we represent all the HJB equation, the QVI HJB equation and the interfere operator as the measure-valued HJB equation in a unified form to establish a unified approach for the optimal control theory of general PDMPs.

英文关键词： Markov Processes;Stochastic Analysis;Generator;Additive functionals;Optimal control