The Study Of The Absolute Ruin And Dividend Payments Problems For Two Risk Models

Risk theory is one of the most important parts in the insurance and actuarialmathematics, and mainly deals with the insurance operation. Ruin theory as the corecontent of the risk theory has been studied over100years. With the deepening researchof risk theory, Hans Gerber and Elias Shiu put forward the concept of Gerber-Shiufunction in1998. As an important risk measure tool, it provides a unified method toanalyze the time of ruin, the surplus immediately before ruin, the deficit at ruin andrelated actuarial quantity, and has been widely used in the study of ruin theory. Similarly,participating insurance is a very important new insurance products, the study ofdividend strategy for participating insurance has drawn more and more actuarialresearchers’ attention. In this thesis, we will study the ruin problems under a generalizedclassical risk model and show how to analyze the Gerber-Shiu function, and study thedividend problems of the perturbed compound Poisson risk model with constant interest,respectively.In the first section, we introduce the classical risk model and its results. Next, wediscuss the various dividend strategys and give the current situation for the study of thedividend strategys. Finally, we give the definition of the Gerber-Shiu functon under theclassical risk model and its introduction.In the second section, we study the absolute ruin probability in the compoundPoisson model with credit and debit interests and liquid reserves. At first, we derive asystem of integro-differential equations with certain boundary conditions for theGerber-Shiu function. Then, applying these results, we obtain asymptotical formula ofthe absolute ruin probability for subexponentially claims. Furthermore, when the claimsare exponentially distributed, we get the explicit expressions for the Gerber-Shiufunction and the exact solution for the absolute ruin probability. Finally, we discuss theGerber-Shiu function when debit interest is varying. In the case of exponentialindividual claim, we give the explicit expressions for the Gerber-Shiu function.In the third section, we consider the compound Poisson risk model perturbed bydiffusion with constant interest and multi-layer dividend strategy. First, the piecewiseintegro-differential equations for the moment-generation function and the nth momentof the present value of all dividends until ruin are derived. Second, we discuss theGerber-Shiu function for which the piecewise integro-differential and integral equations are given. Finally, we get some analytic expressions for the Gerber-Shiu function inspecial cases.