The Maximum Surplus before Ruin and Related Problems in a Jump-Diffusion Renewal Risk Process  

作　　者：Shan Shan WANG Chun Sheng ZHANG 

机构地区：[1]Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, P. R. China [2]School of Mathematics and LPMC, Nankai University, Tianjin 300071, P. R. China

出　　处：《Acta Mathematica Sinica,English Series》2011年第12期2379-2394,共16页数学学报（英文版）

基　　金：Supported by National Basic Research Program of China （973 Program） 2007CB814905, National Natural Science Foundation of China （Grant No. 10871102）, and the Keygrant Project of Chinese Ministry of Education （Grant No. 309009）

摘　　要：In this paper, we investigate a Sparre Andersen risk model perturbed by diffusion with phase-type inter-claim times. We mainly study the distribution of maximum surplus prior to ruin. A matrix form of integro-differential equation for this quantity is derived, and its solution can be expressed as a linear combination of particular solutions of the corresponding homogeneous integro-differential equations. By using the divided differences technique and nonnegative real part roots of Lundberg＇s equation, the explicit Laplace transforms of particular solutions are obtained. Specially, we can deduce closed-form results as long as the individual claim size is rationally distributed. We also give a concise matrix expression for the expected discounted dividend payments under a barrier dividend strategy. Finally, we give some examples to present our main results.In this paper, we investigate a Sparre Andersen risk model perturbed by diffusion with phase-type inter-claim times. We mainly study the distribution of maximum surplus prior to ruin. A matrix form of integro-differential equation for this quantity is derived, and its solution can be expressed as a linear combination of particular solutions of the corresponding homogeneous integro-differential equations. By using the divided differences technique and nonnegative real part roots of Lundberg＇s equation, the explicit Laplace transforms of particular solutions are obtained. Specially, we can deduce closed-form results as long as the individual claim size is rationally distributed. We also give a concise matrix expression for the expected discounted dividend payments under a barrier dividend strategy. Finally, we give some examples to present our main results.

关 键 词：Sparre Andersen risk model phase-type inter-claim times maximum surplus before ruin expected present value of dividends barrier dividend strategy diffusion integro-differential equation 

分 类 号：O211.67[理学—概率论与数理统计] O175.6[理学—数学]