The portfolio model which is proposed by Markowitz is the cornerstone of modern investment theory. The research on portfolio with Markov switching parameters is the main content of this paper. First of all, we discussed the discrete time LQG complete state information with Markov switching parameters. Secondly, the optimal control strategy of the stochastic LQ control with Markov swiching parameters and its applications are provided in the following part. Additionally, the portfolio of the small investors in a competitive state, under the favorable and unfavorable conditions, is researched in this paper, and the optimal investment portfolio strategy is also provided. Finally, this article involved the conditional ruin probability with Markov switching parameters as well, and we made a further research on it under the condition of the stochastic interest rate.This thesis includes six chapters as follows:(1) In chapter1, we summarized significance of the research and a brief introduction of the main results in this thesis is given out and so on.(2) In chapter2, we give out the main tools, lemmas and definitions involved in this paper.(3) In chapter3, considering the influence exerted by regime-switching on the state of the model, Markov chain is introduced into the original model of the discrete time LQG full state information. Utilizing Behrman dynamic programming method, the optimal control strategy is obtained.(4) In chapter4, considering the influence of the state in the classical stochastic LQ control model, this paper extends the model to a jump-diffusion model, meanwhile, a jump-diffusion stochastic Riccati equation and a continuous-time stationary Markov chain are introduced. Then, by applying random variational method, the optimal feedback control of the model is obtained. As applications of the stochastic LQ control framework with Markov chain in the finance market, two examples ware given out in the last part of this chapter.(5) In chapter5, in the financial market, we discussed the investment strategy selection problem under a special condition that there are only two sties securities:one risk securities, and the other risk free securities. Under this condition, both small investors (A and B) should invest on risk free securities and one style risk securities. Under the favorable and unfavorable conditions, introducing Markov chain into the risk price expression, we studied the portfolio selection model and obtained the optimal portfolio strategy.(6) In chapter6, we introduce a Markov chain and extend the Reserve processes to a jump-diffusion model to study the ruin probabilities. By using formula and Martingale method, a partial differential equation satisfied by the finite time horizon conditional ruin probability is obtained. After that, we change the interest in the previous model into a random process and restudy the ruin probabilities.
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