The Gerber-Shiu Function With Multi-layer Strategy In The Phase-type Risk Model Perturbed By Diffusion

Risk theory plays an important part in actuarial mathematics. It has drawn more andmore attention. Because Gerber-Shiu function in the ruin theory is a tool,which includes thedeficit at ruin, the surplus immediately before ruin and ruin probability, and their relationg-ship. Finally, many ruin related problems can be reduced to the study of the Gerber-Shiufunction.In the first part of the paper, we introduce some simple risk models and the Gerber-Shiu function. In the second part, we introduce the Gerber-Shiu function with multi-layerdividend strategy in the classical risk model. The third part is the most important part ofthe paper. In the basis of prevous study, we consider a perturted renewal risk model withmulti-layer dividend strategy in which the inter-claim times are phase-type distributed andthe premium rate is a step function which depends on the current surplus level. We obtainthe systerm of the integro-diferential equations for the Gerber-Shiu function. If the claimamount distribution is the rational family, we obtain the explicit expression of Gerber-Shiufunction. All in all, we provide a recursive approach calculate Gerber-Shiu functions in theabove mentioned risk model.In the classical risk model, such model facilitates the study but may have some difer-ences in the real situations. More and more generalized risk models have be mentioned. Inthe paper, the risk model is multi-layer and has the perturbed term. Because it is diferentfrom the previous models, the integro-diferential equations for the Gerber-Shiu function andcalculation have more complicated changes. In a general way, the defective renewal equation is the common method to study Gerber-Shiu function. Now, we assume the claim amoun-t distribution belongs to the rational family. By the theory of integro-diferential equationand partial franction decomposition, we not only find a good method, but also get satisfyingresult.