Risk Models With Claims By Birth And Death Process

Risk theory is the general theory to quantitatively analyze and predict risks. In the category of actuarial mathematics, bankruptcy theory is the core content of risk theory. The classical bankruptcy theory originated from the Lundberg-Cramer classical risk model. Along with the development of modern insurance, the classical bankruptcy theory cannot have satisfied the requirement of risk management security analysis. Contemporary researchers have made extensions for classical bankruptcy theory and obtained the corresponding results from several aspects, including the fully discrete classical risk model, the bankruptcy theory with heavy-tailed distributions, the bankruptcy theory with composite assets, and the multi-type insurance risk model, etc.In this paper, some extensions for classical risk model have been made. The main research contents include that the claims occurrence process has been extended from Poisson process to pure birth process or birth and death process, and the risk model has been extended from single-type insurance to multi-type insurance.In the first chapter, we review the history and development of bankruptcy theory briefly; introduce the Cramer-Lundberg classical risk models and the main research directions of bankruptcy theory at present.In the second chapter, we construct a risk model with claims by pure birth process which is the generalization of the Poisson classical risk model. A differential integral equation satisfied by the conditional survival probability sequences is obtained. The special cases of tail-Poisson risk model and n-alternate risk model are studied and the ruin probabilities of the exponentially distributed claims or mixed exponentially distributed claims are got.In the third chapter, we construct a risk model with claims by birth and death process. A differential integral equation satisfied by the conditional survival probability sequences is obtained. According to the init1al stationary distribution of birth and death process, an integral equation satisfied by the ruin probability is derived. The upper bound of the rate of convergence of ruin probability is got by generalized renewal method.In the fourth chapter, we construct a multi-type insurance risk model with claims by different birth and death processes. The integral equation satisfied by the ruin probability of bi-type insurance risk model has been studied. The upper bound and lower bound of the rate of convergence of ruin probability are got. The relevant results about bi-type insurance risk model have been generalized to multi-type insurance risk model.