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In this paper, we study the Gerber-Shiu functions for a risk model with two independent classes of risks. We suppose that both of the two claim number processes are renewal processes with phase-type inter-claim times. By re-composing and analyzing the Markov chains associated with two given phase-type distributions, we obtain systems of integro-differential equations for two types of Gerber-Shiu functions. Explicit expressions for the Laplace transforms of the two types of Gerber-Shiu functions are established, respectively. And explicit results for the Gerber-Shiu functions are derived when the initial surplus is zero and when the two claim amount distributions are both from the rational family. Finally, an example is considered to illustrate the applicability of our main results. 相似文献
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In this paper, we investigate the Gerber-Shiu discounted penalty function for the surplus process described by a piecewise deterministic Markov process (PDMP). We derive an integral equation for the Gerber-Shiu discounted penalty function, and obtain the exact solution when the initial surplus is zero. Dickson formulae are also generalized to the present surplus process. 相似文献
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In this paper, a risk model where claims arrive according to a Markovian arrival process (MAP) is considered. A generalization of the well-known Gerber-Shiu function is proposed by incorporating the maximum surplus level before ruin into the penalty function. For this wider class of penalty functions, we show that the generalized Gerber-Shiu function can be expressed in terms of the original Gerber-Shiu function (see e.g. [Gerber, Hans U., Shiu, Elias, S.W., 1998. On the time value of ruin. North American Actuarial Journal 2(1), 48-72]) and the Laplace transform of a first passage time which are both readily available. The generalized Gerber-Shiu function is also shown to be closely related to the original Gerber-Shiu function in the same MAP risk model subject to a dividend barrier strategy. The simplest case of a MAP risk model, namely the classical compound Poisson risk model, will be studied in more detail. In particular, the discounted joint density of the surplus prior to ruin, the deficit at ruin and the maximum surplus before ruin is obtained through analytic Laplace transform inversion of a specific generalized Gerber-Shiu function. Numerical illustrations are then examined. 相似文献
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In this paper, we consider a compound Poisson risk model perturbed by a Brownian motion. We construct the bivariate cumulative distribution function of the claim size and interclaim time by Farlie-Gumbel-Morgenstern copula. The integro-differential equations and the Laplace transforms for the Gerber-Shiu functions are obtained. We also show that the Gerber-Shiu functions satisfy some defective renewal equations. For exponential claims, some explicit expressions are obtained, and numerical examples for the ruin probabilities are also given. 相似文献
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复合Poisson-Geometric风险模型Gerber-Shiu折现惩罚函数 总被引:11,自引:0,他引:11
本文研究赔付为复合Poisson-Geometric过程的风险模型,首先得到了Gerber-Shiu折现惩罚期望函数所满足的更新方程,然后在此基础上推导出了破产概率和破产即刻前赢余分布等所满足的更新方程,再运用Laplace方法得出了破产概率的Pollazek-Khinchin公式,最后根据Pollazek-Khinchin公式,直接得出了当索赔分布服从指数分布的情形下破产概率的显示表达式. 相似文献
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In this paper, we consider the Gerber-Shiu expected discounted penalty function for the perturbed compound Poisson risk process with constant force of interest. We decompose the Gerber-Shiu function into two parts: the expected discounted penalty at ruin that is caused by a claim and the expected discounted penalty at ruin due to oscillation. We derive the integral equations and the integro-differential equations for them. By solving the integro-differential equations we get some closed form expressions for the expected discounted penalty functions under certain assumptions. 相似文献