Analysis of the Gerber-Shiu function and dividend barrier problems for a risk process with two classes of claims

In this paper we consider a risk model with two independent classes of insurance risks. We assume that the two independent claim counting processes are, respectively, the Poisson and the generalized Erlang(2) process. We prove that the Gerber-Shiu function satisfies some defective renewal equations. Exact representations for the solutions of these equations are derived through an associated compound geometric distribution and an analytic expression for this quantity is given when the claim severities have rationally distributed Laplace transforms. Further, the same risk model is considered in the presence of a constant dividend barrier. A system of integro-differential equations with certain boundary conditions for the Gerber-Shiu function is derived and solved. Using systems of integro-differential equations for the moment-generating function as well as for the arbitrary moments of the discounted sum of the dividend payments until ruin, a matrix version of the dividends-penalty is derived. An extension to a risk model when the two independent claim counting processes are Poisson and generalized Erlang(ν), respectively, is considered, generalizing the aforementioned results.     

A class of delayed renewal risk processes with a threshold dividend strategy

This paper considers a class of delayed renewal risk processes with a threshold dividend strategy. The main result is an expression of the Gerber-Shiu expected discounted penalty function in the delayed renewal risk model in terms of the corresponding Cerber-Shiu function in the ordinary renewal model. Subsequently, this relationship is considered in more detail in both the stationary renewal risk model and the ruin probability.     

具有常数利率的延迟更新风险模型

考虑常数利率情形下的延迟更新风险过程．得到了该延迟更新风险模型下的Gerber-Shiu期望折现罚金函数的表达式,并得到了常数利率下的一种特殊的延迟更新风险模型的破产概率的显示表达式.     

具有常数红利边界的两类索赔相关风险模型数

研究常数红利边界下两类索赔相关的风险模型,两类索赔计数过程分别为独立的Poisson过程和广义Erlang(2)过程.利用分解Gerber-Shiu函数的方法,得到了Gerber-Shiu函数满足的积分-微分方程、边界条件、解析表达式及两类索赔额均服从指数分布时的破产概率表达式.     

On a compound Poisson risk model with delayed claims and random incomes

The main focus of this paper is to analyze the Gerber-Shiu penalty function of a compound Poisson risk model with delayed claims and random incomes. It is assumed that every main claim will produce a by-claim which can be delayed with a certain probability. We derive the integral equation satisfied by the Gerber-Shiu penalty function. Given that the premium size is exponentially distributed, the explicit expression for the Laplace transform of the Gerber-Shiu penalty function is derived. Finally, when the premium sizes have rational Laplace transforms, we also obtain the Laplace transform of the Gerber-Shiu penalty function.     

The Gerber-Shiu penalty functions for two classes of renewal risk processes

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.     

Gerber-Shiu function of a discrete risk model with and without a constant dividend barrier

We consider the discrete risk model with exponential claim sizes. We derive the finite explicit elementary expression for the joint density function of three characteristics: the time of ruin, the surplus immediately before ruin, and the deficit at ruin. By using the explicit joint density function, we give a concise expression for the Gerber-Shiu function with no dividends. Finally, we obtain an integral equation for the Gerber-Shiu function under the barrier dividend strategy. The solution can be expressed as a combination of the Gerber-Shiu function without dividends and the solution of the corresponding homogeneous integral equation. This latter function is given clearly by means of the Gerber-Shiu function without dividends.     

On the analysis of the Gerber-Shiu discounted penalty function for risk processes with Markovian arrivals

In this paper, we consider an insurance risk model governed by a Markovian arrival claim process and by phase-type distributed claim amounts, which also allows for claim sizes to be correlated with the inter-claim times. A defective renewal equation of matrix form is derived for the Gerber-Shiu discounted penalty function and solved using matrix analytic methods. The use of the busy period distribution for the canonical fluid flow model is a key factor in our analysis, allowing us to obtain an explicit form of the Gerber-Shiu discounted penalty function avoiding thus the use of Lundberg’s fundamental equation roots. As a special case, we derive the triple Laplace transform of the time to ruin, surplus prior to ruin, and deficit at ruin in explicit form, further obtaining the discounted joint and marginal moments of the surplus prior to ruin and the deficit at ruin.     

A generalized penalty function with the maximum surplus prior to ruin in a MAP risk model

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.     

The expected discounted penalty function under a renewal risk model with stochastic income

In this paper, we consider a renewal risk model with stochastic premiums income. We assume that the premium number process and the claim number process are a Poisson process and a generalized Erlang (n) processes, respectively. When the individual stochastic premium sizes are exponentially distributed, the Laplace transform and a defective renewal equation for the Gerber-Shiu discounted penalty function are obtained. Furthermore, the discounted joint distribution of the surplus just before ruin and the deficit at ruin is given. When the claim size distributions belong to the rational family, the explicit expression of the Gerber-Shiu discounted penalty function is derived. Finally, a specific example is provided.     

随机保费模型下绝对破产概率的可微性以及渐近性（英文）

本文研究了具有随机保费收入的风险模型的Gerber-Shiu罚金函数的可微性以及渐近性质,随机保费收入通过一个复合泊松过程刻画.本文得到了Gerber-Shiu函数所满足的积分微分方程,给出了Gerber-Shiu罚金函数二次可微与三次可微的充分条件.当所讨论的罚金函数是三次可微的时候,前述积分微分方程可以转化为一般的常微分方程.利用常微分方程的标准方法,当个体随机保费和随机理赔都是指数分布的时候,得到了绝对破产概率在初始盈余趋向于无穷大时的渐近性质.     

复合二项风险模型下破产概率的Pollazek-Khinchin公式

本文考虑复合二项风险模型破产概率问题,首先通过研究Gerber-Shiu折现惩罚函数,运用概率论的分析方法得到了其所满足的瑕疵更新方程,再结合离散更新方程理论研究了其渐近性质,最后,运用概率母函数的方法得到了与经典的Gramer-Lundberg模型类似的破产概率Pollazek-Khinchin公式．     

Threshold分红策略下带干扰的两类索赔风险模型的Geber-Shiu函数

本文研究了在threshold分红策略下带干扰的两类索赔风险模型的Geber-Shiu函数.这里假设两个索赔计数过程为独立的更新过程,其中一个为Poisson过程另一个为时间间隔服从广义Erlang(2)分布的更新过程.本文得到了threshold分红策略下Gerber-Shiu函数所满足的积分-微分方程及其边界条件....     

On the Gerber-Shiu function for a risk model with multi-layer dividend strategy

In this paper, we present a new approach to the study of the Gerber-Shiu discounted function for the risk model with multi-layer dividend strategy. The formulae for the Gerber-Shiu discounted function and ruin probability were obtained and the special case where the claim size distribution is a combination of exponentials is considered in detail.     

保费随机复合二项模型的Gerber-shiu折现罚金函数

考虑保费随机收取的复合二项模型.得到了其Gerber-shiu折现罚金函数满足的递推公式,瑕疵更新方程及其渐近解,并且通过构造一个相关的复合几何分布函数,得到了这个更新方程的解析解.相应的也得到了一些相关精算量的渐近表示和分布函数,如破产前瞬时盈余分布的渐近解,导致破产的索赔额的分布函数.     

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In this paper, an Erlang(2) risk model with time-dependent claims is studied under a multi-layer dividend strategy. First, some piecewise integro-differential equations with certain boundary conditions for the Gerber-Shiu function are derived. Then, applying these results, some defective renewal equations and explicit expressions for the Gerber-Shiu function are obtained when the joint density of the inter-claim time and claim size belongs to the rational family.     

阈红利边界下理赔时间间隔与理赔额相依的风险模型

考虑阈红利边界下理赌时间间隔与理赔额相依的风险模型.首先给出了该模型的Gerber- Shiu函数满足的积分.微分方程及更新方程,然后利用Laplace变换及复合几何分布函数得到了Gerber-Shiu函数的确切表达式.     

ON THE RENEWAL RISK MODEL WITH INTEREST AND DIVIDEND

We consider that the reserve of an insurance company follows a renewal risk process with interest and dividend. For this risk process, we derive integral equations and exact infinite series expressions for the Cerber-Shiu discounted penalty function. Then we give lower and upper bounds for the ruin probability. Finally, we present exact expressions for the ruin probability in a special case of renewal risk processes.     

Asymptotic aspects of the Gerber-Shiu function in the renewal risk model using Wiener-Hopf factorization and convolution equivalence

We study the asymptotic behavior of the Gerber-Shiu expected discounted penalty function in the renewal risk model. Under the assumption that the claim-size distribution has a convolution-equivalent density function, which allows both heavy-tailed and light-tailed cases, we establish some asymptotic formulas for the Gerber-Shiu function with a fairly general penalty function. These formulas become completely transparent in the compound Poisson risk model or for certain choices of the penalty function in the renewal risk model. A by-product of this work is an extension of the Wiener-Hopf factorization to include the times of ascending and descending ladders in the continuous-time renewal risk model.     

On the renewal risk model with interest and dividend

We consider that the reserve of an insurance company follows a renewal risk process with interest and dividend. For this risk process, we derive integral equations and exact infinite series expressions for the Gerber-Shiu discounted penalty function. Then we give lower and upper bounds for the ruin probability. Finally, we present exact expressions for the ruin probability in a special case of renewal risk processes.