Research On Some Risk Models

Risk theory is the most important component of the actuarial mathematics and has been studied for more than a century. The classical risk model was firstly introduced in1903by the Swedish mathematician Filip Lundberg[77], and was later developed and made mathematically rigorous by the famous Swedish probabilistic Harald Cramer [28]. In the classical risk model, it is assumed that the premium income is collected con-tinuously with positive determine constant rate and the aggregate claim process is a compound Poisson process. Such assumptions facilitate the study but may be inappro-priate in some real-world situations. To avoid the restriction, my doctoral dissertation is mainly focus on several generalized risk models. This Ph.D. thesis is organized as follows.Chapter1is preface. We introduce the classical risk model and some generalized risk models, mathematical preliminaries and main content of this thesis.In Chapter2, we investigate a renewal risk model in which the distribution of the interclaim times is a mixture of two Erlang distributions. Firstly, the Laplace transform and defective renewal equation for the Gerber-Shiu function are derived. Then two asymptotic results for the Gerber-Shiu function are given when the initial surplus tends to infinity for the light-tailed claims and heavy-tailed claims, respectively. Finally, an explicit expression for the Gerber-Shiu function is given.In Chapter3, we study a risk model with two independent classes of risks, in which both of the two claim number processes are renewal processes with phase-type interclaim times. Using a generalized matrix Dickson-Hipp operator, a matrix Volterra integral equation for the Gerber-Shiu function is derived. The analytical solution to the Gerber-Shiu function is also given.In Chapter4, we consider a risk model where the premiums follow a compound Poisson process and the interclaim times follow a generalized Erlang(n) distribution. We derive a defective renewal equation for the Gerber-Shiu function. The asymptotic and explicit results for the Gerber-Shiu function are also discussed.In Chapter5, we study the ruin problem of a renewal risk process with two-sided jumps. Instead of studying the original risk model, we study an alternative risk model with one-sided jumps. We first derive a defective renewal equation for the ruin probabil-ity, then based on the defective renewal equation we give the asymptotic results for the probability of ruin when the claim sizes have a distribution that belongs to S(v) with v≥0.In Chapter6, a perturbed risk process with proportional reinsurance strategy is studied, in which the interarrival times are generalized Erlang(n) distributed. First, we obtain the defective renewal equation for the Gerber-Shiu function. Then the asymptotic results is discussed when the initial surplus tends to infinity. Finally, numerical results are given to analyze the effect of the reinsurance strategy.In Chapter7, we investigate a risk model with tax, where claims arrive according to a Markovian arrivals. The analytical expression for a generalized Gerber-Shiu function is given. Explicit expression for the expected discounted tax payments is also derived.