The Optimal Dividend Control Strategy Of Two Types Of Risk Models

Mathematical tools gained more and more attention in financial engineering. Especially Gerber, H.U et al applied martingale theory and methods to risk theory, making the discipline rapidly developed. Pricing theory also becomes an important theory in Economics. With the financial and insurance market, insurance companies are no longer satisfied with seeking ruin probability, ruin time and several other actuarial diagnostics, turn to make some minimization measures. These are optimal control problems in financial and insurance. Over the past few decades, through the theory of stochastic control, especially by using the method of Hamilton-Jacobi-Bellman (HJB) equation make rapid progress in this area. However, studies on the optimal control risk models, most work has focused on the numerical solution. However, if one pursuit the precision numerical solution then it must be slow its calculation speed, especially in the Monte Carlo method used in a lot of work in the face of a rapidly changing financial environment appeared to be inadequate. In this paper, based on the classical model by adding realistic factors, the two optimal dividend optimal control model and its explicit solution are researched. This paper is divided into two parts, each introduced a model.(1) In the first model, the claim process is approximated by the Brownian motion with drift and the admissible control policy is composed by the three control variables. First, the insurance company adopts a proportional reinsurance policy. Additionally, the insurance company can invest part of its surplus to risky asset. The risky asset is constructed by a geometric Brownian motion. Finally, this paper concerns the optimal problem of maximizing the expected utility of dividend during the life time of the insurance company. Before obtaining the optimal value function and the specific form of the optimal strategy, this paper gives some nature of the risk model, which does not depend on the specific nature of the expression on the solution. This part of the discussion is given in the second chapter.(2) The second part discusses optimal dividend model when premium rate is less than maximum dividend rate. The main problem of this model is to maximize the value function of the insurance company. Through the standard arguments, this paper gives the corresponding HJB equation. By solving the HJB equation, we obtain the explicit expression of value function and control strategies when the claim size is exponential distribution and premium rate less than maximum dividend rate. This part of the discussion is given in the third chapter.Each chapter is divided into several sections, followed by the order of presentation models, HJB equation, the optimal strategy for solving numerical simulation and model summary. One of the most important features of this paper is that we give a very clear optimal solution.