The Study Of Ruin Probabilities On Some Heavy-tailed Risk Models

In actuarial science, some extreme events, such as floods^earthquakes^volcanic eruptions, etc., once they have occurred, they will have serious impact to insurance com-pany. They will cause insurance company operating difficulties. Some even lead to the insurance company bankruptcy. And such events also occurred frequently in recent years. So people pay more and more attention to the risk process with heavy-tailed claims. In view of this, we consider some kinds of heavy-tailed risk models. The main contents of this thesis are listed in the following:1. In the first chapter, the backgrounds of this thesis are at first introduced. Then, we introduce several common heavy-tailed distribution classes. In addition, we give the standard renewal risk model and the definition of the ruin probability. Finally, some basic definitions, theorems and the main results of this thesis are provided,2. In the second chapter, we discuss the ruin probabilities of a bidimensional risk model with investment. We assume that an insurer has two classes of business. The dependence between the two classes is due to the assumption that they share the same claim-number process. We study two types of ruin in the bidimensional framework. Using the martingale technique, we obtain an upper bound of the infinite-time ruin probability. For each type of ruin, we derive an integral-differential equation of the survival probability, and an explicit asymptotic expression for the finite-time ruin probability.3. In the third chapter, we consider the ruin probabilities of a bidimensional risk model with constant interest rate. In this chapter, we study a risk model in which two insurance companies split the amount they pay out of each claim in proportion. We study two types of ruin in the bidimensional framework. For each type of ruin, we derive an integro-differential equation for the survival probability, and an explicit asymptotic expression for the finite-time ruin probability.4. In the fourth chapter, we study the Gerber-Shiu discounted penalty function and ruin probability of a compound Poisson risk model with constant interest. We obtain an integral-differential equation for the Gerber-Shiu discounted penalty function. So the integral-differential equation for the ruin probability is obtained. In view of this, we also derive an explicit asymptotic expression for the ruin probability.5. In the fifth chapter, we investigate the uniform asymptotic for the ruin probability of a risk model with stochastic premiums income. We derive the asymptotic expression for the finite-time and infinite-time ruin probabilities. And the obtained asymptotic formula holds uniformly for all time horizons.6. In the sixth chapter, we consider the ruin probability of a discrete time risk model under stochastic interest rate. Under the assumption that the common distribution of claim sizes belongs to the class D∩L, a simple asymptotic formula for the ultimate ruin probability is derived.