The Investigation Of Portfolio Based On Markov Regime-switching Model

The optimal portfolio selection problem is a kind of important problems in financial mathematics. In consideration of the real financial market, the portfolio with Markov switching parameters is very realistic and important. So the research on portfolio with Markov switching parameters is the main content of this paper. Firstly, the optimal control strategy of the stochastic LQ control with Markov switching parameters and its applications are provided. Secondly, a kind of risk-sensitive optimal control problem motivated by a portfolio choice problem is obtained by extended maximum principle with Markov switching parameters. Finally, a problem about pricing and hedging of a contingent claim in a continuous-time Markov regime-switching model is investigated.This thesis includes six chapters as follows:In chapter1, the significance of this research is summarized, and a brief introduction of the main results in this thesis is given out and so on.In chapter2, the main tools, lemmas and definitions involved in this paper are obtained.In chapter3, Stochastic LQ control model is extended to the model of jump-diffusion process with continuous-time Markov regime-switching. With the borrowing and lending interest rate, the extended stochastic LQ control can be applied to the continuous-time mean-variance investment policy model in an optimal portfolio selection problem. By using a jump-diffusion stochastic Riccati equation and applying stochastic calculus of variations, the efficient portfolio can be obtained.In chapter4, a kind of risk-sensitive optimal control problem motivated by a portfolio choice problem is obtained by extended maximum principle with Markov switching parameters. There is the real exchange risk for the currency deposit, and the currency exchange rate satisfies the Brown motion. And the model is extended to the model of jump-diffusion process with continuous-time Markov regime-switching. Extended maximum principle which is obtained by the classical convex variational technique is used for the result.In chapter5, the probability Q is obtained by the change of probability measure of both Brown motion and continuous-time Markov chain. Under the probability Q, a pricing and hedging of claim in a Markov regime-switching model can be optioned.In chapter6, the main results in this dissertation are summarized and some issues which need further improvement are pointed out.