Optimal Control Strategies In Compound Markov Binomial Risk Model

In this paper, we consider the optimal control strategies for dividend payments and capital injections in compound Markov binomial risk model with penalties for deficits. The paper is divided into four chapters:In the first chapter, we introduce the background and significance of the research. Then we make a brief description of the current situation and the primarily conclusions about the research. Finally, we point out the main work of this paper.As one core of this paper, in the second chapter, we discuss the optimal control strategy. Based on the risk model with some basic assumptions, we make a preliminary analysis of the optimal dividend strategy and the optimal value function. For two cases divided by whether the dividend rates is restricted, we prove the existence and uniqueness of solution of the HJB equation by contraction mapping principle. We obtain an iterative algorithm for the optimal value function and the optimal dividend strategy. Finally, we transform the value function into the corresponding image function to make the optimal dividend strategy be computed more conveniently.The third chapter is another core of this paper. Based on the risk model in the second chapter, we take in the capital injections. We try to find out the optimal control strategies to maximize the cumulative discounted expected value of net income of the shareholders. We establish the HJB equation satisfied by the value function to find out the optimal value function and the optimal strategies. We find that the optimal value function is the unique solution of the HJB equation.In order to be more easy to program for numerical computation, we transform the value function into the corresponding image function. And the equivalence of some conditions about the optimal control strategies, the optimal value function and the optimal image function is proved. We offer a high efficiency algorithm for obtaining the optimal strategies and the optimal value function. We show some properties of the optimal control strategies and the optimal stopping condition.In the end, we provide some numerical results to illustrate the algorithm.The fourth chapter is given for summary and expectation.