Application Of Martingale In Risk Model

Risk model is a stochastic model for income and claim sequence of insurance company. It is basis of insurance production design and theory of insurance management. Different probability model hypothesis to claim sequence result in many risk models. Martingale is a very important stochastic process. It has many excellent properties and has become basis tool for theoretical research of many fields. Since Gerber (1982) [18] introduced" martingale method" to risk theory, researchers have obtained many good results by it. So the main object of this paper is to make use of "martingale methods" to study bounds of ruin probability of all risk models.We arrange paper as following:In Chapter 1, after giving the existing problems, we give the methods which will be used in the study in section 1.1. Furthermore, the basis knowledge needed in the paper is listed in section 1.2.In chapter 2, we show that how to make a useful martingale and summarize the works which obtained by other author by martingale method for classical and renewal risk model.In Chapter 3, under simple conditions, we studied a multi-arrival process renewal risk model and obtained the up-bounds of finite time and infinite time ruin probability by "martingale method". The martingale is found by the property that "the product of two martingale remains a martingale under some simple condition".In Chapter 4, we study the bidimension claim risk model with disturbing. By "martingale method", an up-bound of ruin probability is obtained. The independence between sub-vectors of a claim vector is not needed. The effect of parameters to the bound is studied. Furthermore, the finite time ruin probability is studied under the heavy-tail case too.In Chapter 5, we studied the discounted discrete risk model, when the stochastic discount rate Y₁,..., Y_n,... are independent and P(Y_i＞0)=1 and independent to claim {X_i}, a simple up-bound of infinite time ruin probability is obtained by "martingale method". When interest rate I₁,..., I_n,... is finite state Markov chain and -1＜I_i≤0, an low-bound is obtained by martingale method too. It extended Cai and Dickson(2004) [64] which assume that Y_i=(1+I_i)^-1 and I_i＞0. Furthermore, under condition that I₁,..., I_n,...is infinite state, an up-bound of ruin probability is obtained. Under very general condition, by "local martingale", we also obtained an up-bound of finite time ruin probability which is related to n. The up-bound fit for little u. It extend the application field of the bound only fitting for large u.In chapter 6, we studied the non-homogeneous Markov's renewal risk model. The exponential up-bound of ruin probability is obtained by super-martingale method. Their special model is studied too.In chapter 7, we study a renewal risk model with mixed exponential distribution arrival interval. The expression of Laplace transformation of penalty function was obtained.