Research On Stochastic Differential Game For Investment & Insurance Problems Under Markov Regime-switching Models

Since it was created, differential game has garnered a lot of attention,and over half a century has now become a scientifically effective decision- making tool. The subject of this paper, stochastic differential games for dynamic systems, has been introduced via three examples of the reality. Differential game is a useful tool for dealing with competition when facing social life and financial markets with full of confrontation.This dissertation investigates non cooperative stochastic differential game of dynamic systems that are descripted by It??? stochastic differential equations by using the stochastic optimal control and maximum principle,the direct approach in stochastic differential game theory. The criterions of the existence condition of game equilibrium strategy are given, and the design method of equilibrium strategy are presented , also, the results are applied to the stochastic robust H_?control, stochastic H₂/H_? control and portfolio selection problem and the optimal reinsurance problem of insurance companies. The main contributions can be concluded as follows:First, problems linear stochastic differential systems driven by It??? stochastic differential equation are discussed, a non zero sum differential game problem Nash and zero sum differential game problem are constructed and the explicit expression of Nash equilibrium and saddle point game equilibrium are got in the model of complete information and incomplete information, respectively. In the condition of complete information, the equilibrium strategies are depend on the state and the solution of a set of coupled Riccati equations and backward differential equations; in incomplete information, the optimal filtering and condition estimation error are obtained by taken advantage of Kalman-Bucy filtering theory,and the equilibrium strategies are depend on the optimal filtering of state and the solution of a set of coupled Riccati equations and backward differential equations.Second, problems of the stochastic differential game for linear quadratic systems driven by Poisson jumps diffusion Brownian motion with state- and control-dependent noise were studied, the Nash equilibrium was obtained of this game. Then, we apply the results to modern Robustness control by game theory, and promote it to incomplete information systems by filtering theory. We get the equilibriums and obtain that the existence conditions of Nash equilibrium strategies are equivalent to solution of two cross-coupled matrix Riccati equations , and the existence conditions of saddle point equilibrium are equivalent to solution of a matrix Riccati equation. In incomplete information,equilibriums are depend on the filtering equation of dual state. We verify the application of the results in financial market portfolio optimization problem.Third, we study on the stochastic differential game of Markov jump linear systems .First of all, we discuss the stochastic differential game problem of two players with linear Markov non-singular system. Then,it is extended to the condition of N ?N>2? players with linear Markov singular system, the finite time Nash differential game problem is discussed. The Nash equilibrium is obtained and the existence conditions of Nash equilibrium strategies are equivalent to solution of two cross-coupled matrix Riccati equations. Then, the finite time Nash differential game with N players is extended to infinite time condition, and the Nash equilibrium is obtained,whose existence conditions are equivalent to solution of algebraic Riccati equations. Finally,the theory was applied to H₂/H_? control problem,which extends the application of game theory.Fourth, we investigate the application of the differential game theory in finance and insurance. The problem of portfolio with Markov switching system and the insurance company investment & reinsurance under CEV model are studied via maximum principle.The optimal investment strategy of investor and the optimal investment reinsurance strategy of insurance company are obtained, respectively. Investors can make decisions based on these results.