The Gerber-Shiu Expected Discounted Penalty Function For Risk Processes Under Absolute Ruin

As an important element in actuarial science, risk theory has attracted the attentionfrom actuarial scholars. With the widely applications of probability theory, stochasticprocesses in actuarial science. Ruin theory, as an important branch of risk theory, itsresearch has been developing rapidly and successfully into practice, and it became animportant basis for the insurance company decision-making and risk control. In recentyears, the study of the surplus of the insurance company under absolute ruin time hasbecome more and more popular. Further, we take Gerber-Shiu expected discountedpenalty function to describe the change of surplus. The Gerber-Shiu expecteddiscounted penalty function at ruin being a powerful analytical tool made it possible toanalyze the time of ruin, the surplus immediately before ruin, the deficit at ruin, andrelated quantities in a unified manner. The purpose of the study for expected discountedpenalty function at absolute ruin time, is to get better control for funds and avoidbankruptcy. In this paper, content arranged as follows.The first chapter is devoted to the summary of the dissertation, which recounts thedevelopment and the present circumstance about the related problems. Specifically, thecurrent development of the risk process, the classical risk model and its mainconclusions, and the expected discounted penalty function at ruin time. Moreover, themain results of the present paper are stated.In Chapter2, we consider the absolute ruin in compound Poisson risk model withtwo nonnegative interests and a constant dividend barrier. First, an auxiliary function isdefined for the Gerber-Shiu expected discounted penalty function. Second, anintegro-differential equation for the auxiliary function is derived. Finally, in the case ofexponential individual claim, the explicit expressions for the Gerber-Shiu expecteddiscounted penalty function are obtained by the auxiliary function. In the end, we obtainthe Laplace transform of the time of ruin.In Chapter3, the main job is similar as the second chapter. The difference is thatwe have several barriers, they are all constant, between different barriers, the interestsare different, and we obtain the integro-differential equation for the auxiliary function.In the case of exponential individual claim, we derive the analytical expression for theexpected discounted penalty function at ruin, the absolute ruin probability, the n-thmoment of the deficit at absolute ruin, and the Laplace transform of the deficit prior absolute ruin.In Chapter4, on the basis of the second and third chapter, we make the risk modelexpand to a multi-layered situation, that means, we have many constant barriers, also,the interests are different, and derived some simple expressions for the auxiliaryfunction. Finally, we get some prospect, such as a risk model with linear dividendbarrier or nonlinear dividend barrier, etc.