The Probability Of Ruin And Optimal Dividend Problem In Several Risk Models

Recently, dividend optimization problems for financial and insurance corporations have attracted extensive attention. How should corporation pay dividends to its share-holders? A possible goal is that the company tries to maximize the expectation of the discounted dividends until possible ruin of the company. This optimization prob-lem was first proposed by De Finetti (1957). In recent years, quite a few interesting papers deal with the optimal dividend problem. There are two types of dividend poli-cies:barrier strategy and threshold strategy are extensively studied. The theory and method have been employed in the present paper, such as renewal theory, probabil-ity theory, stochastic control and martingale theory to investigate the dividend and optimal dividend problems.The dissertation is divided into six chapters.In Chapter â… , the two types risk models and the background of dividend problems and some basic concepts and theorems have been introduced.In Chapter â…¡, the optimal dividend problem for a spectrally positive Levy pro-cess has been investigated. We study the optimal dividend problem for a company whose surplus process evolves as a spectrally positive Levy process before dividends are deducted. We assume that dividends are paid to the shareholders according to an admissible strategy whose dividend rate is bounded by a constant. The integro-differential equations with boundary conditions satisfied by the expectation of the sum of discounted dividends until ruin is derived, then we get the exact expression. We show that the optimal dividend strategy is formed by a threshold strategy.In Chapter â…¢, the optimal dividend problem with a terminal value in spectrally positive Levy processes has been studied. Using the fluctuation theory of spectrally positive Levy processes we give an explicit expression of the value function of a barrier strategy. Subsequently we show that a barrier strategy is the optimal strategy among all admissible ones.In Chapter IV, the dividend in a dual risk model with a barrier dividend strategy has been dealt with. We derive integro-differential equations with boundary condi-tions satisfied by the expectation of the sum of discounted dividends until ruin and the moment-generating function of the discounted dividend payments until ruin respec-tively. The results are illustrated by several examples.In Chapter â…¤, the first-exit time for jump processes having double-sided jumps with rational Laplace transforms have been studied. We derive the joint distribution of the first passage time to two-sided barriers and the value of process at the first passage time. As applications, we present explicit expressions of the dividend formulae for barrier strategy and threshold strategy.In Chapter â…¥, a general two-sided jump-diffusion risk model that allows for risky investments as well as for correlation between the two Brownian motions driving insur-ance risk and investment return has been focused on. The model has been introduced firstly, then we get the integro-differential equations satisfied by the Gerber-Shiu func-tions as well as the expected discounted penalty functions at ruin; We also study the dividend problem for the threshold and barrier strategies, the moments and moment-generating function of the total discounted dividends until ruin are discussed. Some examples are given for special cases.