The Expected Discounted Penalty Function For Sparre Andersen Risk Model

Risk theory is an important branch of financial insurance and insurance actuarial,its core is theory of ruin.In 1998,Gerber and Shiu proposed the concept of expected discounted penalty function in the classical risk model for the first time.After that,the research on the issue of ruin changed to the research of expected discounted penalty function.Classical risk model is relatively ideal model.With the extensive research of this model,the classical risk model was gradually extended to more general risk models,such as Perturbed risk model,Sparre Andersen risk model,Multiple risk model.Investment return risk model.Omega risk model,etc.Based on the classical risk model,this paper studies the expected discounted penalty function in different situations for the Sparre Andersen risk process.This paper is divided into three parts.The first chapter is the introduction,which mainly introduces the research background of risk theory.First.the classical risk model and its four generalized models which are Perturbed risk model,Sparre Andersen risk model,Multiple risk model and Investment return risk model are introduced.Second,two main concepts in insurance actuarial are introduced,including ruin and bankruptcy,expected discounted penalty function and their research status.Final,it introduces the main work of this paper.The second chapter focuses on the investment return risk model of continuous-time with phase-type inter-claim times and two-side jumps.Firstly,the integro-differential equations that satisfying the expected discounted penalty function are obtained by the full probability formula.Secondly,when the upward jumps have a exponential distribution and the downward jumps have an arbitrary distribution,the explicit,expressions for the expected discounted penalty function are obtained by Laplace transform and other calculation methods.The third chapter bases on the classical risk model,it researches the expected discounted penalty function about bankruptcy,with exponential inter-claim times and mixture of exponential claim sizes.First,the integro-differential equations that satisfying the expected discounted penalty function are obtained by the full probability formula.Then,the differential equations for the expected discounted penalty function are acquired when the claim sizes are mixture of exponential distributions.Last,for the constant bankruptcy rate function,the explicit expressions of the expected discounted penalty function are gained.