Gerber-Shiu Analysis Under Some Risk Models

In recent years, mathematical modelling and risk analysis in risk theory have drawn more and more actuarial researchers'attention. As an important risk measure tool, the Gerber-Shiu function has been widely used in the study of ruin theory, and accordingly, many ruin related problems can be reduced to the evaluation of the Gerber-Shiu function. In this thesis, we will study the ruin problems under some risk models, and show how to analyze the Gerber-Shiu function.Firstly, two classes of perturbed Sparre Andersen risk models are respectively considered in Section 2 and Section 3, where the perturbation motions are respectively spectrally negative Lévy process and jump-diffusion process with double-exponential jumps. We derive the integro-differential equations for the Gerber-Shiu functions by using some potential measures, and then determine the Laplace transforms and defective renewal equations for the Gerber-Shiu functions. Some asymptotic results are also obatined when the claim size distribution is heavy-tailed.Secondly, we study two classes of risk models with multi-layer dividend strategy. In Section 4, we consider a Sparre Andersen risk model with generalized Erlang(n) interclaim times. We distinguish the cases with and without Brownian motion perturbation, and in both cases we show how to calculate the Gerber-Shiu functions. While in Section 5, a compound Poisson risk model with constant investment interest force and debit interest force is considered. Piecewise integro-differential equations for the Gerber-Shiu function are derived, and some explicit expressions and asymptotic results are also given for exponential claim size distribution and subexponential claim size distribution, respectively.Thirdly, we investigate the evaluation of the Gerber-Shiu functions in two classes of risk models with two-sided jumps. In Section 6, we consider a continuous time risk model and study a generalized Gerber-Shiu function by using some techniques in renewal theory. While in Section 9, we consider a discrete time risk model and show how to find the generating function and the recursive formula for the Gerber-Shiu function.Finally, we study the Gerber-Shiu functions in the Markovian additive risk models. In Section 7, we consider a continuous time risk model, where the company is allowed to borrow money to continue its operation when the surplus becomes negative. In this model, the integro-differential equations for the Gerber-Shiu functions are derived and the solutions are discussed. Some asymptotic results are also given when the claim sizes distributions are heavy-tailed. While in Section 8, we consider a discrete time risk model, and give a recursive approach to calculate the Gerber-Shiu function..