On The Risk Model Involving Two Classes Of Claims With Threshold Dividend Strategy

In this thesis, we consider a risk model with two independent classes of insurancerisks. The expected discounted penalty (Gerber-Shiu ) function and the expectationof the present value of all dividends until ruin (the value function) for the risk modelon a threshold dividend strategy are studied. The thesis is divided into 5 chapters.In chapter 1, the classical compound Poisson risk model, the risk model in-volving two independent classes of insurance risks and two core problems in theactuarial literature, the expected discounted penalty function and dividend strategyare introduced. Some corresponding developments in recent years are also reviewed.In chapter 2, we consider a double compound Poisson risk model involving twoindependent Poisson processes. A system of integro-differential equations of theexpected discounted penalty function is derived. We also analyze the expected dis-counted penalty function. When 0≤u < b, the Laplace transform method is appliedand consequently the explicit expression for the expected discounted penalty func-tion is derived when the claims from the two classes are exponentially distributed.When u≥b, a system of the renewal equations for the expected discounted penaltyfunction is obtained.In chapter 3, we study the value function in the double compound Poissonrisk model with the threshold dividend strategy. Integro-differential equations withcertain boundary conditions for the present value of dividends until ruin are de-rived. When the claim size distributions are exponentially distributed, we obtainthe explicit expressions for the value function and show that the threshold dividendstrategy is an optimal dividend strategy.In chapter 4, a risk model involving Poisson and Erlang(n) risk processes isstudied. We derive a system of integro-differential equations of the expected dis-counted penalty function. The Laplace transform method and the renewal equationmethod are applied to analyze the expected discounted penalty function.In chapter 5, we study the value function in the risk model involving Poisson andErlang(n) processes. Integro-differential equations with certain boundary conditionsfor the present value of dividends until ruin are derived, the explicit expression ofthe value function and the problem of the optimal dividend strategy should bereserved for further consideration, since it is diffcult to solve the corresponding integro-differential equations.