Stochastic Risk Models In Actuarial Theory

This paper has investigated the ruin problem of several stochastic risk models in actu-arial theory by the renewal argument, martingale theory, Markov process, stochastic anal-ysis and Laplace transform. The Lundberg upper bound of ruin probability, the property of Gerber-Shiu expected discounted penalty function are discussed. The main ideas and contributions of this thesis are as follows:1. Describing the disturbance of the stochastic factors by Brownian Motion, we con-sider the renewal risk model perturbed by diffusion. Making use of the stationary and inde-pendent characters of the jump-diffusion risk model, we get the general formula and Lund-berg's inequality of the ruin probability. Gerber-Shiu expected discounted penalty function in the diffusion-perturbed Erlang(2) risk model is discussed. Using certain mathematical technique, we derive the integro-differential equation of the Gerber-Shiu function.2. The renewal risk models in a Markovian environment are investigated. Firstly, in the Markov-modulated jump-diffusion risk model, we obtain the Volterra integral equations for the ruin probabilities through Laplace transform. Secondly, considering the Markov-modulated Erlang(2) risk model, we obtain the Lundberg upper bound of the ruin proba-bilities by the martingale technique, and the integral equations of Gerber-Shiu functions by the backward differential argument.3. We introduce a dependence structure between the claim size and the inter-claim time to Erlang(2) risk model, namely, the FGM Copula. In this framework, we study the property of the generalized Lundberg's fundamental equation's roots by the Rouche's the-orem, and obtain the integro-differential equation of the Gerber-Shiu function.