Research On Proportional Reinsurance Problems With Several Continuous-time Risk Model

This paper mainly research the optimal control problem with several types of risk model by using the theory of stochastic control and dynamic program-ming principle. We mainly discuss the classical jump-diffusion risk model and a bivariate reserve risk process the optimal proportional reinsurance risk model and investment strategy problem when the insurer is allowed to purchase rein-surance and invest in one risk-free asset and one risk asset whose price process satisfies the Heston model. The main work of this paper is as follows:In Chapter1, Simple summarizes the research background and the latest research in the theory of risk dynamically. Then, we present the main work of this paper and the main result of my research.In Chapter2, we mainly introduces several kinds of classical risk model and the market investment price process will be used in this article.In Chapter 3, this part we studies the optimal reinsurance and investmen-t issues with a jump-diffusion risk model. In view of the two explanations of the diffusion terms in the model, namely "U-C situation" and "A-C situation", assumed that the instantaneous rate of investment return follows a Heston mod-el. Proportional reinsurance dynamic strategy and variance premium principle were adopted in this part. The objective of the insurer is to maximize the ex-pected utility of terminal wealth. Using stochastic control theory we obtain the Hamilton-Jacobi-Bellman equations which satiated by the value function, ex-plicit expressions for the optimal strategy and value function are derived with two kinds of cases, also compares the differences between them. At the same time in "A-C situation", for example, we present a numerical example to illus-trate the effects of various parameters of the model on the optimal reinsurance and investment strategy.In Chapter 4, this chapter as two related insurance business the back-ground, we study the compound Poisson risk model of binary dependent claims under the optimal proportional reinsurance and investment problem. The in-surance company use expect premium principle calculate the premium income, assumed that the instantaneous rate of investment return follows a Heston mod-el. Using the dynamic programming principle with the goal of maximize the expected utility, we prove the existence and uniqueness of the optimal propor-tional reinsurance strategy. Also we find the optimal reinsurance strategy im-plicit function expression and the explicit expression for the optimal investment strategy and value function.In Chapter 5, on the basis of compound Poisson risk model in binary depen-dent claims, we get the surplus Poisson approximation model after a standard Brownian motion diffusion approximation. For the insurance company long-term interests, we explore the smallest ruin probability of two-dimensional rein-surance strategy Dynamic Settings. Through solving the HJB equation which satisfied by two related variables, we get the explicit solution of the optimal reinsurance strategy.