Markovian Modulation Risk Model With Perturbation Under Constant Interest Rate

In 1905, Lundberg and Cramer proposed the compound Poisson risk model. It is also called the classical risk model in insurance theory, where premiums are regarded as profits and claims are viewed as costs. But actually it’s too idealized. It could not be satisfied in the real world. So many academics devote to the generalization of the classical risk model in different ways. The model modulated by the Markov chain is a generalization of the classical risk model. The model is also called the Markovian regime switching risk model in finance and actuarial science literature. This model can capture the feature that insurance policies may need to change when economical or political environment changes.In 1998, Gerber and Shiu proposed the expected discounted penalty function which is commonly referred to as Gerber-Shiu function in the ruin literature. The study of the function has been a hot topic in actuarial science. The function has a great many of advantages. It has attracted more attention in recent years, which provides a unified treatment of the time of ruin, the surplus immediately before ruin and the deficit at ruin.In this paper, we study the generalized expected discounted penalty function in a risk process with interest and diffusion.In this paper, we study the Markov-modulated risk model with interest perturbed by diffusion and dividend payments with a threshold strategy in the Markov-modulated risk model with interest perturbed by diffusion.The body of the paper includes three chapters:In the first chapter, we give the descriptions, basic assumptions of the classical risk model, definition of the ruin time and the probability of ruin. It also introduces the basic knowledge of continuous-time Markov chain;In the second chapter, we consider the Markov-modulated risk model with interest perturbed by diffusion and the integro-differential equations satisfied by the generalized Gerber-Shiu function are given. We also study its Laplace transform and the exit times;At last chapter, we consider the dividend payments with a threshold strategy in the Markov-modulated risk model with interest perturbed by diffusion. We obtain the integro-differential equations satisfied by the expected discounted dividend function, the moment-generating function and the n th moments of the discounted dividend payments.