导　　师： 张成科

学科专业： 1201

授予学位： 博士

作　　者： ;

机构地区： 广东工业大学

摘　　要： 自1965年Rufus· Isaacs出版了第一部微分博弈专著《Differential Games》以来,无论其理论还是应用研究都得到了很大的发展,今天,微分博弈已被广泛应用于国防军事工程、生产管理、经济生活等领域的各个方面,成为了科学有效的决策工具。本学位论文以工程和经济领域中大量存在的一类动态系统/(工程领域称之为Markov切换系统,经济管理学界称之为Markov调制系统,本论文统称为Markov切换系统/)为研究对象,在已有Markov切换系统最优控制理论和随机微分博弈理论的基础上,利用动态优化理论中的极大值原理、动态规划原理、Riccati方程法等,系统研究线性Markov切换系统的非合作随机微分博弈理论,并给出其在均值-方差型投资组合选择和保险公司投资-再保险问题中的应用分析。主要的研究结果如下： 一、研究了噪声仅依赖于状态的线性Markov切换系统、目标泛函为正定二次型的随机微分博弈问题,称之为正定型线性Markov切换系统的随机微分博弈。首先,在已有随机线性二次/(linear quadratic, LQ/)微分博弈理论的基础上,建立了线性Markov切换系统二人零和博弈和非零和博弈模型。然后借助于随机LQ控制中的均方稳定的概念,给出并证明了系统均衡策略存在的充要条件等价于相应的广义矩阵Riccati方程存在解,同时得到了最优控制策略的显式解和最优值函数的表达式。最后在此基础上将所得结果应用于线性Markov切换系统的随机H∞、H2／H∞控制上,并给出了数值仿真算例验证结果的正确性,拓展了已有的随机微分博弈的相关研究成果。 二、研究了噪声同时依赖于状态和控制的线性Markov切换系统、目标泛函为不定二次型的随机微分博弈问题,称之为线性Markov切换系统的不定随机微分博弈。首先,借助于随机不定LQ控制中的相关结果,建立了线性Markov切换系统二人零和及非� Since Rufus· Isaacs published his first monograph 'Differential Games' in1965, great development has been made about its theory and application. Today, differential game has been widely used in many aspects, such as national defense and military engineering, production management, economic life, etc, and it has been a scientific and effective tool for decision-making. This dissertation investigated a class of dynamic systems which have been used frequently in engineering and economics /(engineering experts referred them as the Markov jump systems, economic and management scholars referred them as the Markovian regime-switching systems, in this dissertation, they are collectively referred to the Markov jump systems/). On the basis of the existing literature of optimal control for Markov jump systems and stochastic differential game theory, by utilizing the maximum principle, dynamic programming, Riccati equation methods used in dynamic optimization, this dissertation studied the stochastic differential game theory of Markov jump linear systems and its applications in finance and insurance systematically. The main contributions can be concluded as follows: First, problems of stochastic differential game for Markov jump linear systems with state-dependent noise were discussed, we called them definite stochastic differential games for Markov jump linear systems. Firstly, on the basis of the existed stochastic LQ differential game theory, two person zero-sum and nonzero-sum game models of Markov jump linear systems were established. And then by means of the concept of mean-square stabilizability in stochastic LQ control, we proved that necessary and sufficient conditions for the existence of the equilibrium strategy are equivalent to the solvability of the corresponding generalized matrix-valued Riccati equations; moreover, we got the explicit solution of the optimal control strategy and the expressions of the optimal value function. Finally, on the basis of the obtained results, we investigated the stochastic H�

关 键 词： 线性 切换系统 随机微分博弈 随机 控制 调制模型 投资组合 投资 再保险

领　　域： [经济管理] [理学] [理学]