Research On Two Types Optimal Problems In A BMAP Model

In this article, we use probability and statistics, stochastic processes, Markov decision theory, stochastic control theory, and dynamic programming principle to study the optimal investment problem with transaction costs in a MAP(Markov Arrival Process) model, the other is the optimal dividend and capital injection problem of a company in a BMAP (Batch Markov Arrival Process) model. The main ideas of these are as follows:In Chapter 1, we introduce the background of the risk theory and its de-velopment. Then, we present the main work of this paper and the main results of my research.In Chapter 2, we mainly introduce preliminary of BMAP model and some risk relevant models.In Chapter 3, we consider the optimal investment problem with transaction costs in a MAP model. We allow the investor to invest his wealth in a financial market consisting of a risk-free asset and a risky asset. The price processes of the risk-free asset and the risky asset are modulated by the phase process of a MAP, which is an observable continuous-time Markov chain. The possible investments are restricted to some random discrete time points which are determined by the same MAP. The investor has investment opportunities at some of these time points and can do nothing at the remaining random time points. Then, we con-struct an auxiliary Markov chain, obtain the recursion of the value functions and the exact form of the optimal investment strategies under the assumption that interarrival times obeys exponential distribution and the terminal wealth is evaluated by the power utility. Additionally, we present a numerical example to analyze the influence of the economic conditions, the investor’s investment opportunities and the transaction costs on investor’s terminal utility and in-vestment strategies.In Chapter 4, we discuss the optimal dividend and capital injection prob- lem of a company in a BMAP model. The parameters in the process of the company’s surplus are modulated by the phase process of the BMAP, which is an observable continuous-time Markov chain. The possible dividend and capital injection are restricted to some random discrete time points which are determined by the same BMAP. The company has both dividend and capital injection opportunities or only has dividend but not capital injection opportu-nities at some of these time points, while can do nothing at other random time points. By transforming the BMAP model to an auxiliary Markov modulat-ed model, we study the optimal dividend and capital injection problem of the company under the assumption that the company won’t bankrupt. This chapter aims to maximize the difference between the total expected discounted dividend and the amount of capital and obtain the exact solution of the value functions and the optimal dividend and capital injection strategy. Finally, we present a numerical example to analyze the influence of the economic conditions on the company’s value functions and dividend and capital injection strategies.