Discussion Of The Bankruptcy Of Several Types Of Risk Models

In the insurance mathematics category, the risk theory is its important constituent, it mainly handles the problems of the ruin probability of the stochastic risk model in the insurance business. The research of ruin theory begins from Sweden actuary Fillip Lundberg, who proposed the Lundberg-Cramer risk model. Afterwards, Hans U. Gerber becomes the leading scholar who does research in ruin theory at present. He introduced martingale method into the research of ruin theory, he also deepened the research of the classical theory. His book〈An Introduction to Mathematical Risk Theory〉(Chinese translation) [1] has become the classical works in this area now. One big contributions of Gerber is introduce the Brownian motion into classical risk model. J. Grandell's monograph (Aspects of Risk Theory) [2] is also an important literature in this area. In the point process frame, it extensions the total claim amount process. It discussed the situation when the claim counting process follows the renewal process and the Cox process, respectively. This thesis will unify these two points, discusses a jump-diffusion renewal process, then we obtain an asymptotic formula for the finite-time ruin probability under the assumptions that the claim-sizes are heavy-tailed distributed. In the classical risk theory, independence of risks is often assumed. However, due to the increasing complexity of insurance and reinsurance products, actuaries have recently paid more attention to the model of dependent risks. In which the research of multi-dimensional risk model is extremely complex. Even in the two-dimensional case, the problem is also challenging. This thesis will discuss the ruin probability of a two-dimensional risk model with positive and negative risk sums.The thesis is divided into three sections according to the contents. In Chapter 1, we give an overview to ruin theory. The accurate descriptions, fundamental assumption and main results of Lundberg-Cramer risk model are given. Two modern approaches in researches of ruin theory: Feller's renewal argument and Gerber's martingale method, are introduced and used to prove the conclusions of classical risk model concisely. Then the other advances in modern ruin theory are also briefly mentioned.In Chapter 2, we consider the finite-time ruin probability for the jump-diffusion renewal process. Under the assumptions that the claim-sizes are sub-exponentially distributed and that the interest force is constant, we obtain an asymptotic formula for the finite-time ruin probability. Then we contrast the result with the situation when the interest force is 0.In Chapter 3, we consider a bivariate risk model with positive and negative risk sums. Three different types of ruin probabilities are defined. Using some results of one-dimensional risk process, simple bounds for the two-dimensional ruin probabilities are obtained. We derive the integral-differential equation for the ruin probabilityφ_a(a₁u₁+ a₂u₂). An integral-differential equation of survival probabilityφ_min(u₁,u₂) is derived by renewal method. At last we get its Laplace transform.