线性Markov切换系统的随机微分博弈理论及在金融保险中的应用研究
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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零和不定随机微分博弈模型,推导证明了随机微分博弈问题适定及均衡策略存在的充要条件等价于相应的矩阵Riccati微分(代数)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。最后给出数值仿真算例验证了所得结果的有效性,同时也为后面章节的研究奠定了基础。
三、基于博弈方法研究了的线性Markov切换系统目标泛函不定的鲁棒控制问题。借助于线性Markov切换系统不定随机微分博弈的结果,将控制策略设计者视为博弈的一方即博弈人P1,将随机性干扰视为博弈的另一方即“自然博弈人”P2,从而将鲁棒控制问题转化为两人博弈问题,即博弈人P1如何在预期到“自然人”P2的各种干扰策略情况下设计自己的策略,既实现与“自然人”的均衡又使自己的目标最优。解决了噪声同时依赖于状态、控制和干扰的线性Markov切换系统的随机H∞、H2／H∞混合控制问题,证明了控制器的存在性,并借助耦合Riccati微分(代数)方程给出了反馈增益明晰的表达式,最后给出数值算例验证了所得结论的有效性。
四、研究了线性Markov切换系统微分博弈理论在金融保险中的应用。运用随机微分博弈的方法讨论了基于Markov调制模型的均值.方差型投资组合选择问题。首先假设资产价格服从带Markov调制的几何布朗运动,建立了带Markov调制的金融市场模型,然后将市场看成“虚拟”的博弈对手,在投资者与市场之间构建了一个二人零和随机微分博弈模型,投资者选择一个投资策略最大化其终止时刻财富期望效用,而市场选择一个概率测度代表的投资“环境”最小化投资者的最大化终止时刻期望财寓效用。最后在投资者具有常数相对风险规避系数效用函数偏好的假设下,通过求解微分博弈问题对应的HJBI方程,得到了投资者的最优投资策略及最优值函数的显式解。
接着,讨论了基于Markov调制模型的保险公司投资-再保险问题。首先假设保险公司的盈余过程是一个带Markov调制的随机过程,金融资产的价格服从带Markov调制的几何布朗运动,建立了带Markov调制的金融市场模型,然后将市场看成“虚拟”的博弈对手,在保险公司与市场自然之间构建了一个二人零和随机微分博弈模型,保险公司选择一个投资-再保险策略最大化其终止时刻财富期望效用,而市场选择一个概率测度代表的经济“环境”最小化保险公司的最大效用。通过使用动态规划的方法求得了问题的最优解,同时通过验证定理给出了问题的HJB解,最后在适当的假设条件下给出了保险公司和市场最优策略的显式解及最优函数值。
本论文的研究得到国家自然科学基金项目—广义随机线性Markov切换系统非合作微分博弈理论及其在金融保险中的应用(71171061)和广东省自然科学基是金—随机Markov切换系统的非合作微分博弈理论及在经济中的应用(S2011010000473)的支持。
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∞, H2/H∞control problems for Markov jump linear systems by applying the game theory approach, and got the optimal control strategy. These obtained results in this chapter expanded the existing results in stochastic differential game research.
Second, problems of the stochastic differential game for Markov jump linear systems with state-and control-dependent noise were studied, similar with indefinite stochastic LQ problems, we called them indefinite stochastic differential game for Markov jump linear systems. Firstly, two person zero-sum and nonzero-sum game models of Markov jump linear systems are formulated by applying the results of stochastic LQ problems. Then, we proved that the well-posedness and the existence condition of the equilibrium strategy are equivalent to the solvability of the corresponding matrix-valued differential (algebraic) Riccati equations; meanwhile, the explicit solution of optimal control strategy and the expression of the optimal value function were obtained. Finally, numerical simulation examples were given to verify the validity of the presented results, and laid the foundation for the sequel chapters.
Third, we studied the robust control problems of Markov jump linear systems based on game theory approach. By means of the results of indefinite stochastic differential game for Markov jump linear systems discussed in chapter4, we viewed the control strategy designer as one player of the game, i.e. P1, the stochastic disturbance as another player of the game, i.e."nature" P2, respectively, the robust control problems were transformed into a two person differential game model, player P1faced the problem that how to design his own strategy in the case of various interference strategy implemented by "nature" P2, both balanced with the "nature" and optimized his own objective. Corresponding results of stochastic H∞H2/H∞control problems for Markov jump linear systems with state, control and disturbance-dependent noise were obtained, and proved the existence of the controller, clarity expressions of the feedback gain were given by means of coupled differential (algebraic) Riccati equations. Finally, numerical examples were presented to verify the validity of the conclusions.
Fourth, we investigated the application of the differential game theory of Markov jump linear systems in finance and insurance. We studied a game theoretic approach for portfolio selection under Markovian regime-switching models. First, we considered the portfolio selection problem in a Markovian regime switching Black-Scholes economy to illustrate the main idea of the method. In this case, the price dynamics of the underlying risky asset is governed by a Markovian regime switching geometric Brownian motion (GBM). Then, we considered the portfolio selection problem in the context of a two-player, zero-sum stochastic differential game. One of the players in this game is an investor and the other is a fictitious player-the market. The investor has a CRRA utility function and is to select a portfolio, which maximizes the expected CRRA utility of the terminal wealth. The market then selects a generalized "scenario", which is represented by a probability measure, to minimize the maximal utility of the investor. The closed-form expressions of optimal strategies of the investor and the optimal value function are derived by solving the HJBI equation of the associated game.
Then, we studied a game theoretic approach for optimal investment-reinsurance problem of an insurance company under Markovian regime-switching models. Firstly, we considered the optimal investment-reinsurance problem in a Markovian regime switching Black-Scholes economy. In this case, the price dynamics of the underlying risky asset is governed by a Markovian regime switching geometric Brownian motion (GBM). Then, we considered the problem in the context of a two-player, zero-sum stochastic differential game. One of the players in this game is an insurance company and the other is a fictitious player-the market. The insurance company has a utility function and is to select an investment-reinsurance policy, which maximizes the expected utility of the terminal wealth. The market then selects a generalized "scenario", which is represented by a probability measure, to minimize the maximal utility of the insurance company. The optimal solutions were given by applying the dynamic programming; and meanwhile, the HJB solutions of the associated game were obtained. Finally, under appropriate assumptions, the closed-form expressions of optimal investment-reinsurance policy of the insurance company and the optimal value function were derived.
This thesis was supported by the National Natural Science Fund of China—Noncooperative differential game theory of generalized Markov jump linear systems with application to finance and insurance (71171061), the Natural Science Fund of Guangdong Province—Noncooperative differential game theory of Markov jump linear systems with application to economics (S2011010004970).
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零和不定随机微分博弈模型,推导证明了随机微分博弈问题适定及均衡策略存在的充要条件等价于相应的矩阵Riccati微分(代数)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。最后给出数值仿真算例验证了所得结果的有效性,同时也为后面章节的研究奠定了基础。
三、基于博弈方法研究了的线性Markov切换系统目标泛函不定的鲁棒控制问题。借助于线性Markov切换系统不定随机微分博弈的结果,将控制策略设计者视为博弈的一方即博弈人P1,将随机性干扰视为博弈的另一方即“自然博弈人”P2,从而将鲁棒控制问题转化为两人博弈问题,即博弈人P1如何在预期到“自然人”P2的各种干扰策略情况下设计自己的策略,既实现与“自然人”的均衡又使自己的目标最优。解决了噪声同时依赖于状态、控制和干扰的线性Markov切换系统的随机H∞、H2／H∞混合控制问题,证明了控制器的存在性,并借助耦合Riccati微分(代数)方程给出了反馈增益明晰的表达式,最后给出数值算例验证了所得结论的有效性。
四、研究了线性Markov切换系统微分博弈理论在金融保险中的应用。运用随机微分博弈的方法讨论了基于Markov调制模型的均值.方差型投资组合选择问题。首先假设资产价格服从带Markov调制的几何布朗运动,建立了带Markov调制的金融市场模型,然后将市场看成“虚拟”的博弈对手,在投资者与市场之间构建了一个二人零和随机微分博弈模型,投资者选择一个投资策略最大化其终止时刻财富期望效用,而市场选择一个概率测度代表的投资“环境”最小化投资者的最大化终止时刻期望财寓效用。最后在投资者具有常数相对风险规避系数效用函数偏好的假设下,通过求解微分博弈问题对应的HJBI方程,得到了投资者的最优投资策略及最优值函数的显式解。
接着,讨论了基于Markov调制模型的保险公司投资-再保险问题。首先假设保险公司的盈余过程是一个带Markov调制的随机过程,金融资产的价格服从带Markov调制的几何布朗运动,建立了带Markov调制的金融市场模型,然后将市场看成“虚拟”的博弈对手,在保险公司与市场自然之间构建了一个二人零和随机微分博弈模型,保险公司选择一个投资-再保险策略最大化其终止时刻财富期望效用,而市场选择一个概率测度代表的经济“环境”最小化保险公司的最大效用。通过使用动态规划的方法求得了问题的最优解,同时通过验证定理给出了问题的HJB解,最后在适当的假设条件下给出了保险公司和市场最优策略的显式解及最优函数值。
本论文的研究得到国家自然科学基金项目—广义随机线性Markov切换系统非合作微分博弈理论及其在金融保险中的应用(71171061)和广东省自然科学基是金—随机Markov切换系统的非合作微分博弈理论及在经济中的应用(S2011010000473)的支持。
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∞, H2/H∞control problems for Markov jump linear systems by applying the game theory approach, and got the optimal control strategy. These obtained results in this chapter expanded the existing results in stochastic differential game research.
Second, problems of the stochastic differential game for Markov jump linear systems with state-and control-dependent noise were studied, similar with indefinite stochastic LQ problems, we called them indefinite stochastic differential game for Markov jump linear systems. Firstly, two person zero-sum and nonzero-sum game models of Markov jump linear systems are formulated by applying the results of stochastic LQ problems. Then, we proved that the well-posedness and the existence condition of the equilibrium strategy are equivalent to the solvability of the corresponding matrix-valued differential (algebraic) Riccati equations; meanwhile, the explicit solution of optimal control strategy and the expression of the optimal value function were obtained. Finally, numerical simulation examples were given to verify the validity of the presented results, and laid the foundation for the sequel chapters.
Third, we studied the robust control problems of Markov jump linear systems based on game theory approach. By means of the results of indefinite stochastic differential game for Markov jump linear systems discussed in chapter4, we viewed the control strategy designer as one player of the game, i.e. P1, the stochastic disturbance as another player of the game, i.e."nature" P2, respectively, the robust control problems were transformed into a two person differential game model, player P1faced the problem that how to design his own strategy in the case of various interference strategy implemented by "nature" P2, both balanced with the "nature" and optimized his own objective. Corresponding results of stochastic H∞H2/H∞control problems for Markov jump linear systems with state, control and disturbance-dependent noise were obtained, and proved the existence of the controller, clarity expressions of the feedback gain were given by means of coupled differential (algebraic) Riccati equations. Finally, numerical examples were presented to verify the validity of the conclusions.
Fourth, we investigated the application of the differential game theory of Markov jump linear systems in finance and insurance. We studied a game theoretic approach for portfolio selection under Markovian regime-switching models. First, we considered the portfolio selection problem in a Markovian regime switching Black-Scholes economy to illustrate the main idea of the method. In this case, the price dynamics of the underlying risky asset is governed by a Markovian regime switching geometric Brownian motion (GBM). Then, we considered the portfolio selection problem in the context of a two-player, zero-sum stochastic differential game. One of the players in this game is an investor and the other is a fictitious player-the market. The investor has a CRRA utility function and is to select a portfolio, which maximizes the expected CRRA utility of the terminal wealth. The market then selects a generalized "scenario", which is represented by a probability measure, to minimize the maximal utility of the investor. The closed-form expressions of optimal strategies of the investor and the optimal value function are derived by solving the HJBI equation of the associated game.
Then, we studied a game theoretic approach for optimal investment-reinsurance problem of an insurance company under Markovian regime-switching models. Firstly, we considered the optimal investment-reinsurance problem in a Markovian regime switching Black-Scholes economy. In this case, the price dynamics of the underlying risky asset is governed by a Markovian regime switching geometric Brownian motion (GBM). Then, we considered the problem in the context of a two-player, zero-sum stochastic differential game. One of the players in this game is an insurance company and the other is a fictitious player-the market. The insurance company has a utility function and is to select an investment-reinsurance policy, which maximizes the expected utility of the terminal wealth. The market then selects a generalized "scenario", which is represented by a probability measure, to minimize the maximal utility of the insurance company. The optimal solutions were given by applying the dynamic programming; and meanwhile, the HJB solutions of the associated game were obtained. Finally, under appropriate assumptions, the closed-form expressions of optimal investment-reinsurance policy of the insurance company and the optimal value function were derived.
This thesis was supported by the National Natural Science Fund of China—Noncooperative differential game theory of generalized Markov jump linear systems with application to finance and insurance (71171061), the Natural Science Fund of Guangdong Province—Noncooperative differential game theory of Markov jump linear systems with application to economics (S2011010004970).
引文
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