The Sparre Andersen Risk Process with Investment and Debit Interest

In this paper, we study absolute ruin problems for the Sparre Andersen risk process with generalized Erlang()-distributed inter-claim times, investment and debit interest. We first give a system of integro-differential equations with certain boundary conditions satisfied by the expected discounted penalty function at absolute ruin. Second, we obtain a defective renewal equation under some special cases, then based on the defective renewal equation we derive two asymptotic results for the expected discounted penalty function when the initial surplus tends to infinity for the light-tailed claims and heavy-tailed claims, respectively. Finally, we investigate some explicit solutions and numerical results for generalized Erlang(2) inter-claim times and exponential claims.     

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.     

带干扰广义Elang(n)风险过程的破产前最大盈余

本文考虑了索赔时间间距为广义Erlang(n)分布的带干扰更新(Sparre Andersen)风险过程.所用的方法类似于Albrecher,et al.(2005),即将广义Erlang(n)随机变量分解成n个独立的指数随机变量的和.建立了破产前最大盈余所满足的积分-微分方程,讨论了索赔量分布为K<,m>分布时的特殊情形.     

The expectation of aggregate discounted dividends for a Sparre Anderson risk process perturbed by diffusion

In this paper, we study the expectation of aggregate dividends until ruin for a Sparre Andersen risk process perturbed by diffusion under a threshold strategy, in which claim waiting times have a common generalized Erlang(n) distribution. For this strategy, we assume that if the surplus is above certain threshold level before ruin, dividends are continuously paid at a constant rate that does not exceed the premium rate, and if not, no dividends are paid. We obtain some integro-differential equations satisfied by the expected discounted dividends, and further its renewal equations. Finally, applying these results to the Erlang(2) risk model perturbed by diffusion, where claims have a common exponential distributions, we give some explicit expressions and numerical analysis. Copyright © 2007 John Wiley & Sons, Ltd.     

Surplus analysis for a class of Coxian interclaim time distributions with applications to mixed Erlang claim amounts

Gerber-Shiu analysis with the generalized penalty function proposed by Cheung et al. (in press-a) is considered in the Sparre Andersen risk model with a K_n family distribution for the interclaim time. A defective renewal equation and its solution for the present Gerber-Shiu function are derived, and their forms are natural for analysis which jointly involves the time of ruin and the surplus immediately prior to ruin. The results are then used to find explicit expressions for various defective joint and marginal densities, including those involving the claim causing ruin and the last interclaim time before ruin. The case with mixed Erlang claim amounts is considered in some detail.     

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.     

一类混合分红策略下的广义Erlang(n)风险模型

本文考虑混合分红策略下索赔来到间隔为广义Erlang(n)分布的更新风险模型,利用指数分布的无记忆性,分别得到破产前期望折现分红函数和折现分红的矩母函数满足的积分-微分方程及其边界条件.最后给出索赔为指数分布及索赔来到间隔为广义Erlang(2)分布的风险模型的期望折现分红函数的精确表达式.     

具有二步保费的Erlang（2）风险模型

本文考虑了当索赔间隔时间为Erlang(2)分布且保费收取为二步保费过程的复合更新风险模型,推导出该模型的罚金折现期望值函数满足具有一定边界条件和积分微分方程,并解出该方程.特别地,当索赔额为指数分布时,利用所得结果给出了破产时间的Laplace变换及终积破产概率的解析解.     

A class of Sparre Andersen risk process

In this paper, we investigate a renewal risk model in which the distribution of the interclaim times is a mixture of two Erlang distributions. First, the Laplace transform and the defective renewal equation for the Gerber-Shiu function are derived. Then, two asymptotic results for the Laplace transform of the time of ruin are given when the initial surplus tends to infinity for the light-tailed claims and heavy-tailed claims, respectively. Finally, an explicit expression for the Gerber-Shiu function is given.     

The Gerber-Shiu function and the generalized Cramér-Lundberg model

In this paper we extend some results in Cramér [7] by considering the expected discounted penalty function as a generalization of the infinite time ruin probability. We consider his ruin theory model that allows the claim sizes to take positive as well as negative values. Depending on the sign of these amounts, they are interpreted either as claims made by insureds or as income from deceased annuitants, respectively. We then demonstrate that when the events’ arrival process is a renewal process, the Gerber-Shiu function satisfies a defective renewal equation. Subsequently, we consider some special cases such as when claims have exponential distribution or the arrival process is a compound Poisson process and annuity-related income has Erlang(n, β) distribution. We are then able to specify the parameter and the functions involved in the above-mentioned defective renewal equation.     

LAPLACE TRANSFORM OF THE SURVIVAL PROBABILITY UNDER SPARRE ANDERSEN MODEL

In this paper a class of risk processes in which claims occur as a renewal process is studied. A clear expression for Laplace transform of the survival probability is well given when the claim amount distribution is Erlang distribution or mixed Erlang distribution. The expressions for moments of the time to ruin with the model above are given.     

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??The paper considers a risk model with two dependent classes of
insurance business. In this model, the two claim number processes are partly sparsely
correlated through an Erlang(2) process. By introducing an auxiliary model, we obtain the
integral equations for ultimate ruin probabilities, and discuss the asymptotic property of
ruin probabilities by renewal approach. We also get the linear differential equations of
ruin probabilities of the model and the corresponding auxiliary model when claims follow
the exponential distributions, and show how solves the linear differential equations by a
specific example.     

Ruin probabilities in the discrete time renewal risk model

In this paper, we study the discrete time renewal risk model, an extension to Gerber’s compound binomial model. Under the framework of this extension, we study the aggregate claim amount process and both finite-time and infinite-time ruin probabilities. For completeness, we derive an upper bound and an asymptotic expression for the infinite-time ruin probabilities in this risk model. Also, we demonstrate that the proposed extension can be used to approximate the continuous time renewal risk model (also known as the Sparre Andersen risk model) as Gerber’s compound binomial model has been proposed as a discrete-time version of the classical compound Poisson risk model. This allows us to derive both numerical upper and lower bounds for the infinite-time ruin probabilities defined in the continuous time risk model from their equivalents under the discrete time renewal risk model. Finally, the numerical algorithm proposed to compute infinite-time ruin probabilities in the discrete time renewal risk model is also applied in some of its extensions.     

The Perturbed Compound Poisson Risk Process with Investment and Debit Interest

In this paper, we study absolute ruin questions for the perturbed compound Poisson risk process with investment and debit interests by the expected discounted penalty function at absolute ruin, which provides a unified means of studying the joint distribution of the absolute ruin time, the surplus immediately prior to absolute ruin time and the deficit at absolute ruin time. We first consider the stochastic Dirichlet problem and from which we derive a system of integro-differential equations and the boundary conditions satisfied by the function. Second, we derive the integral equations and a defective renewal equation under some special cases, then based on the defective renewal equation we give two asymptotic results for the expected discounted penalty function when the initial surplus tends to infinity for the light-tailed claims and heavy-tailed claims, respectively. Finally, we investigate some explicit solutions and numerical results when claim sizes are exponentially distributed.     

一类相依两险种风险模型的分类破产概率

本文考虑文[1]中引入的一类索赔达到计数过程相关的两险种风险模型.利用更新方法,获得了该风险模型的分类破产概率的渐进结果,并给出了指数索赔情形下分类破产概率的表达式,从而改进了文[1]中的相关结果.     

Ruin probability and time of ruin with a proportional reinsurance threshold strategy

In this paper, we present a threshold proportional reinsurance strategy and we analyze the effect on some solvency measures: ruin probability and time of ruin. This dynamic reinsurance strategy assumes a retention level that is not constant and depends on the level of the surplus. In a model with inter-occurrence times being generalized Erlang(n)-distributed, we obtain the integro-differential equation for the Gerber?CShiu function. Then, we present the solution for inter-occurrence times exponentially distributed and claim amount phase-type(N). Some examples for exponential and phase-type(2) claim amount are presented. Finally, we show some comparisons between threshold reinsurance and proportional reinsurance.     

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

Quantities of interest in ruin theory are investigated under the general framework of the expected discounted penalty function, assuming a risk model where both premiums and claims follow compound Poisson processes. Both a defective renewal equation and an integral equation satisfied by the expected discounted penalty function are established. Some implications that these equations have on particular quantities such as the discounted deficit and the probability of ultimate ruin are illustrated. Finally, the case when premiums have Erlang(n,β) distribution and the distribution of the claims is arbitrary is investigated in more depth. Throughout the paper specific examples where claims and premiums have particular distributions are provided.     

一类具有新红利界限的Erlang(2)风险模型

考虑到保险公司的实际运作中红利的发放率要比保费的收取率小,将一类新的红利政策引入Erlang(2)风险模型,利用更新论证,得到并求解了此模型下罚金折现期望函数所满足的微积分方程.最后通过数值例子,分析了红利界限与初始盈余对破产概率的影响.     

On a perturbed Sparre Andersen risk model with multi-layer dividend strategy

In this paper, we consider a perturbed Sparre Andersen risk model, in which the inter-claim times are generalized Erlang(n) distributed. Under the multi-layer dividend strategy, piece-wise integro-differential equations for the discounted penalty functions are derived, and a recursive approach is applied to express the solutions. A numerical example to calculate the ruin probabilities is given to illustrate the solution procedure.     

The Asymptotic Estimate of Absolute Ruin Probabilities in the Renewal Risk Model with Constant Force of Interest

In this paper, absolute ruin problems for a kind of renewal risk model with constant interest force are studied. For certain situations of the claim distribution with heavy tail, consider the surplus of the arrival time, and discrete the surplus process, then use the method of renewal function and convolution, we present the asymptotic properties of absolute ruin probability when the initial surplus tends to infinity.