Ruin Probability In Generalized Risk Models

Risk theory is the analysis of the stochastic features of insurance business. In the classical risk model, the number of claims from an insurance portfolio is assumed to follow a Poisson process, the individual claim sizes are independent and identical random variables, and the premiums are described by a constant rate of income. In this kind of model, Filip Lundberg and Harald Cramer obtain the expression and estimate of ruin probability. Furthermore, William Feller get the Lundberg-Cramer-type estimate for the ruin probability by the aid of a renewal technique. Hans Gerber get the exponential upper bounds for the ruin probability using a martingale approach.The problem on the severity of ruin has recently received a remarkable attention. A series of risk model which can better describe the practical operation of insurance company has been extended by many scholars. There are several common extension:1. Generalization of the claim-arrival process.Under the assumption that the claim-arrival process is the renewal process, Cox process, generalized compound Poisson process, Gamma process and inverse Gaussian process etc., the asymptotic ruin probability is obtained.2. Generalization of the premium-arrival process.Similar to the generalization of the claim-arrival process, the premium-arrival process also can be extended to Poisson process, Cox process and renewal process etc. Furthermore, the insurance premium is not the constant but random variables.3. In order to describe the reality of insurance business, the interest force, stochastic interfere and dividends are considered.4. Generalize the ruin probability from the continuous risk model to the discrete risk model.This thesis is devoted to a study of ruin probability. By the means of stochastic process theory, we introduce several generalized risk models. In detail, three aspects of content are considered.1. We get a risk model of thinning process perturbed by diffusion, where the arrival of term policies follows a Poisson process and the arrival of claims follows ap-thinning process of the arrival process. We get Lundberg inequality and formula of the ruin probability by martingale approach. We prove that the Lundberg exponent of our model is larger than that of the classical risk model when the claims exponentially distributed.2. We discuss the modified classical risk model perturbed by diffusion with a linear dividend barrier. We get the ruin probability and derive the integral-differential equation satisfied by the survival probability. When the level of the dividend barrier approaches the limit constant premium, we find that the classical risk model perturbed by diffusion is a limit of our model.3. Under the discrete time, we discuss a new model which the arrival of premium policies and the occurrence of claims follow two compound binomial processes. The Lundberg inequality is obtained by two approach. We prove the integral equation of ruin probability. As an application, we give the counting method of ruin probability for the case of exponential distribution.