Research On Related Issues Of Bilateral Jump Risk Model Under Three Bonus Strategies

In the classical risk model,for the convenience of calculation,premiums are charged at a constant rate,w,hile claims are expressed by downward jumps.Sometimes the company may have random incomes,for instance,the pension funds that the insurer paid to its policyholders can partly become the random income when a policyholder dies.The two-sided jumps risk model describes both the claims and the random incomes.Many researches have been done since the proposal of the two-sided jumps risk model.More attention has been paid to study the Gerber-Shiu function with no dividend barrier,while few articles have covered the dividend strategies.Partly because the way to analyse the quantities involving the dividend barrier is to solve the related differential equation.This method has its limitations when the upward jump is added.In the classical risk model with dividend barrier,the general solution to the differential equation for the Gerber-Shiu function can be expressed as a linear combination of the Gerber-Shiu function with no barrier and the solutions to theassociated homogeneous integro-diferential equation.Following this idea,we rewrite the integro-differential equation for the Gerber-Shiu function in the two-sided jumps risk model with dividend barrier into the higher-order integro-differential equation,in order to eliminate the integral term which contains the probability density function of the upward jump sizes.Then,we can analyse the Gerber-shiu function by using the approach used in the classical risk model with dividend barrier.The same idea is also applicable to the analysis of the mth moment function of the discounted sum of dividend payments until ruin in the two-sided jumps risk model with dividend barrier.In this thesis,we study the two-sided jumps risk model under three dividend barrier strategies,in which we assume that the distribution of the upward jump sizes is specific and the downward jump sizes follow an arbitrary distribution.Under the constant barrier dividend strategy,an integro-differential equation with boundary conditions for the expected discounted penalty function is derived and the solution is provided.The defective renew-al equation for the expected discounted penalty function with no barrier,as well as the integro-differential equation for the mth moment function of the discounted sum of dividend payments until ruin,are derived.When the dividend barrier is a linear dividend barrier,the integro-differential equation with boundary conditions for the expected discounted penalty function and the m.th moment function of the discounted sum of dividend payments until ruin are derived,respectively.Under the threshold strategy,an integro-differential equation with boundary conditions for the expected discounted penalty function is derived and the solution is provided.The defective renewal equation for the expected discounted penalty function with no barrier is derived.We also give an example to obtain the expression of the expected discounted penalty function and the mth moment function of the discounted sum of dividend payments until ruin,when the distribution of the claim amounts is specific.In Chapter 5,on the basis of the two-sided jumps risk model under a constant barrier dividend strategy,we add another random incomes.We assume that both distributions of the upward jump sizes are different exponential distributions,and the downward jump sizes follow an arbitrary distribution.The significance of this model is that,in reality,insurance companies may have different types of insurance.Each type of the insurance will bring different random incomes due to its different nature,that is,the arrival process of its random income may be different.We also give some examples to obtain the expression of Mb(u)and mth moment function of the discounted sum of dividend payments until ruin,when the distribution of the claim amounts is specific.