Research On Stochastic Differential Game Theory For Markov Jump Linear Systems With Applicaton To Fianance And Insurance

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).