连续时间马氏决策过程均值-方差优化问题的研究

项目名称： 连续时间马氏决策过程均值-方差优化问题的研究

项目编号： No.11201182

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： 叶柳儿

作者单位： 暨南大学

项目金额： 22万元

中文摘要： 本项目主要研究连续时间马氏决策过程的均值-方差优化问题。拟解决以下三个问题：1）针对Markowitz均值-方差模型，在期望折扣收益最大化或等于某个给定常数的前提下，寻找相应方差最小的策略。通过分析其与折扣准则的理论关系，得到均值-方差最优策略存在的条件，进而得到其计算方法；2）通过建立受约束连续时间MDP均值-方差模型，在期望收益不小于给定常数的条件下，寻找使方差达到最小的策略。运用受约束模型的已有理论结果，分析当前准则下最优策略的存在性以及它的计算方法；3）利用风险中立动态规划新方法，处理连续时间MDP中的折扣、平均和均值-方差最优化问题，建立最优策略存在性，进一步分析相应的计算方法。另外，将分析该方法与现有理论方法的区别和联系，从而扩大MDP的应用范围。以上三个问题的研究均是首次的。

中文关键词： 均值-方差准则；受约束的马氏决策过程；Markov对策；最优性条件；最优策略

英文摘要： In this project, we consider a series of mean-variance optimality problems for continuous-time Markov decision processes (MDPs). The main object is to obtain some policies that minimize the variance over a set of all policies with a given expected reward, which satisfies suitable conditions. We are planning to solve the three following questions: 1) For Markowitz mean-variance models, we aim to find a policy that minimizes the variance over a set of all policies with a optimal/given expected reward. Using the conditional expectation and Markov property we can prove that the mean-variance optimality problem can be transformed to an equivalent discounted optimality problem, and establish the existence of mean-variance optimal policies. Furthermore, we will analysis their computational methods. 2) We establish the constrained continuous-time MDPs models with mean-variance optimality criterion. Using the theory of constrained continuous-time MDPs, we will find the condition of existence of constrained mean-variance optimal policies and their computational methods. 3) We first introduce the concept of risk-averse dynamic programming, and employ the Markov risk measures. Using these new tools，we will establish the existence of discounted/average/mean-variance optimal policies. Moreover, we will analysis the differenc

英文关键词： Mean-variance criterion；Constrained Markov decision processes；Markov games；Optimality conditions；Optimal policy