Research On Ruin Problems And Optimal Control Problems For Several Classes Risk Models

This paper investigate the ruin problems and the optimal control prob-lems for several classes risk models by the renewal argument, stochastic control theory, Markov process and martingale theory. The risk model that we study can roughly be divided into two kinds:one is the discrete time risk model with random income, the other is the jump diffusion risk model (or the compound Poisson risk model perturbed by diffusion). The main ideas and contributions of this thesis is organized as follows.1. In chapter2, we extend the deterministic premiums income in the clas-sical discrete time compound binomial risk model to the binomial process, and suppose that the claims maybe delayed. That is to say, we consider the com-pound binomial risk model with delayed claims and random income. Making use of the character of the stationary and independent increments of the risk process and the approach of generating function, under the constant dividend barrier, we obtain the explicit formula for the expected present value of total dividend payments prior to ruin. In addition, the impacts of the initial capi-tal and the delay of by-claims on the expected present value of total dividend payments prior to ruin are discussed by two examples.2. In chapter3, based on the model in the chapter2, we extend the as-sumption that each main claim induces a by-claim with certainly in the most of the previous literature to the case that each main claim causes a by-claim with a certain probability. That is to say, we consider a compound binomial risk model with correlated claims and random income, and study the expected present value of total dividend payments prior to ruin in this model. Fortu-nately, the conclusion of this chapter can include the result as in chapter2. Finally, the influence of the parameters in this model on the expected present value of total dividend payments prior to ruin are discussed by two examples, respectively. 3. In chapter4, depth study on the model in the chapter3, almost all risk models described in the most of the previous literature relied on the assumption that the force of interest or the discount factor per period is a constant, in this chapter, we use a time-homogeneous Markov chain with a finite state space to model the one-period interest rates. That is to say, we consider a compound binomial risk model with correlated claims-, random income and stochastic interest, and study the expected present value of total dividend payments prior to ruin in this model. A general expression for the expected present value of total dividend payments prior to ruin are derived. Explicit results are obtained in two examples.4. In chapter5, based on the classic compound Poisson risk model, and describing the disturbance of the stochastic factors by a stand Brownian mo-tion, double-threshold dividend strategy to shareholders and policy-holders is addressed here. A system of integro-differential equations with certain bound-ary conditions for the expected present value of total dividend payments prior to ruin and Gerber-Shiu expected discounted penalty function is derived and solved. Firstly, we translate the integro-differential equations into the renewal equations that identical to the integro-differential equations, then show that the solutions of the renewal equations are unique. Based on these, by iteration, their closed-form solutions are obtained.5. In chapter6, we consider an optimal investment and proportional rein-surance problem of an insurer whose surplus process follows a jump diffusion model. In our model the insurer transfers part of the risk due to insurance claims via a proportional reinsurance and invests the surplus in a "simplified" financial market consisting of a risk-free asset and a risky asset. The dynamics of the risky asset are governed by a constant elasticity of variance model to incorporate conditional heteroscedasticity. The reinsurance premium is calcu-lated according to the variance principle. The diffusion term can explain the uncertainty associated with the surplus of the insurance company or the ad- ditional small claims. Our objective is to maximize the expected exponential utility of terminal wealth. This optimization problem is studied in two cases depending on the diffusion term’s explanation. In all cases, by using techniques of stochastic control theory, closed-form expressions for the value function and optimal proportional reinsurance and investment policies are obtained.6. In chapter7, we study the optimal investment and proportional reinsur-ance strategy for an insurance company with jump diffusion risk model. The dynamics of the risky asset are governed by a geometric Levy process. Un-der the criterion of maximizing the expected exponential utility from terminal wealth, with a constraint on the proportional reinsurance strategy, closed-form expressions for the value function and optimal strategy are obtained. Numerical examples are presented to show the impact of model parameters on the optimal strategies.7. In chapter8, we discuss an optimal portfolio and reinsurance problem of an insurance company facing model uncertainty via a game theoretic ap-proach. The insurance company invests in a security market described by the Black-Scholes model. The risk process of the company is governed by either a jump-diffusion process or its diffusion approximation. The company can also transfer a certain proportion of the insurance risk to a reinsurance company by purchasing reinsurance. The optimal portfolio and reinsurance problem is formulated as two-player, zero-sum, stochastic differential games between the insurance and the market. We obtain closed-form solutions to the game prob-lems in both the jump-diffusion risk process and its diffusion approximation.