复合Poisson-Geometric风险模型Gerber-Shiu折现惩罚函数

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.     

离散的三项分布风险模型的渐近解

本文研究了离散的三项分布风险模型,在调节系数存在的前提下,借助于离散更新方程的一个极限定理,对于充分大的初始盈余给出了最终破产概率、破产前一刻的盈余和破产时赤字的概率的渐近解．其结果可以在离散的多项分布风险模型中得到推广．     

复合Poisson-geometric风险模型下第n次索赔时的破产概率研究

在复合Poisson-geometric风险模型下,通过构造一个特殊的Gerber-Shiu函数,推导出此风险模型下Gerber-Shiu函数满足的更新方程,破产时刻和直到破产时的索赔次数的联合密度函数,得到了第n次索赔时的破产概率的数学表达式．     

On the discounted penalty function in the discrete time stationary renewal risk model

In this paper we consider the discrete time stationary renewal risk model. We express the Gerber-Shiu discounted penalty function in the stationary renewal risk model in terms of the corresponding Gerber-Shiu function in the ordinary model. In particular, we obtain a defective renewal equation for the probability generating function of ruin time. The solution of the renewal equation is then given. The explicit formulas for the discounted survival distribution of the deficit at ruin are also derived.     

保费到达为平衡更新过程的复合更新风险模型

本文研究保费到达为平衡更新过程的复合更新风险模型 ,给出了有限时间内的生存概率分布 ,破产时间 T与破产时资产盈余 U(T)的联合分布 ,及破产时间 T与破产前瞬时盈余 U(T- )的联合分布 .     

带常利率的Erlang(2)风险模型的罚金折现期望函数

The purpose of this paper is to consider the expected value of a discounted penalty due at ruin in the Erlang（2） risk process under constant interest force. An integro-differential equation satisfied by the expected value and a second-order differential equation for the Laplace transform of the expected value are derived. In addition, the paper will present the recursive algorithm for the joint distribution of the surplus immediately before ruin and the deficit at ruin. Finally, by the differential equation, the defective renewal equation and the explicit expression for the expected value are given in the interest-free case.     

On the expected discounted penalty function for the compound Poisson risk model with delayed claims

In this paper, we consider an extension to the compound Poisson risk model for which the occurrence of the claim may be delayed. Two kinds of dependent claims, main claims and by-claims, are defined, where every by-claim is induced by the main claim and may be delayed with a certain probability. Both the expected discounted penalty functions with zero initial surplus and the Laplace transforms of the expected discounted penalty functions are obtained from an integro-differential equations system. We prove that the expected discounted penalty function satisfies a defective renewal equation. An exact representation for the solution of this equation is derived through an associated compound geometric distribution, and an analytic expression for this quantity is given for when the claim amounts from both classes are exponentially distributed. Moreover, the closed form expressions for the ruin probability and the distribution function of the surplus before ruin are obtained. We prove that the ruin probability for this risk model decreases as the probability of the delay of by-claims increases. Finally, numerical results are also provided to illustrate the applicability of our main result and the impact of the delay of by-claims on the expected discounted penalty functions.     

复合二项风险模型下Gerber-Shiu折现惩罚函数的渐近解

首先研究了二项风险模型下Gerber-Shiu折现惩罚函数所满足的瑕疵更新方程,然后根据离散更新方程理论研究了其渐近解,并得到了破产概率、破产即刻前赢余和破产时刻赤字的联合分布分布以及其边际分布等的渐近解,进一步完善了Pavlova K P和Willmot G E 2004年发表的相关问题的结果.     

多重延迟的关键更新定理及其在风险理论中的应用

利用广义局部次指数分布族的性质,讨论了带有多重延迟且Lundberg指数不存在时的关键更新定理,所得结果包含了重尾和轻尾的情形.将此结果应用到平稳更新风险模型,得到了该模型在破产时亏损额分布的局部渐近性质.     

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.     

On a risk model with random incomes and dependence between claim sizes and claim intervals

In this paper, we construct a risk model with a dependence setting where there exists a specific structure among the time between two claim occurrences, premium sizes and claim sizes. Given that the premium size is exponentially distributed, both the Laplace transforms and defective renewal equations for the expected discounted penalty functions are obtained. Exact representations for the solutions of the defective renewal equations are derived through an associated compound geometric distribution. When the claims are subexponentially distributed, the asymptotic formulae for ruin probabilities are obtained. Finally, when the individual premium sizes have rational Laplace transforms, the Laplace transforms for the expected discounted penalty functions are obtained.     

The compound binomial risk model with time-correlated claims

In this paper, we consider the compound binomial risk model with the time-correlated claims. It is assumed that every main claim will produce a by-claim but the occurrence of the by-claim may be delayed. We obtain the recursive formula of the joint distribution of the surplus immediately prior to ruin and deficit at ruin. Furthermore, the ruin probability is given by means of ruin probability and the deficit at ruin of the classical compound binomial risk model. Finally, we derive an upper bound for the ruin probability.     

Analysis of risk models using a level crossing technique

This paper analyzes ruin-like risk models in Insurance, which are variants of the Cramer–Lundberg (C–L) model with a barrier or a threshold. We consider three model variants, which have different portfolio strategies when the risk reserve reaches the barrier or exceeds the threshold. In these models we construct a time-extended risk process defined on cycles of a specific renewal process. The time until ruin is equal to one cycle of the specific renewal process. We also consider a fourth model, which is a variant of a model proposed by Dickson and Waters (2004). The analysis of each model employs a level crossing method (LC) to derive the steady-state probability distribution of the time-extended risk process. From the derived distribution we compute the expected time until ruin, the probability distribution of the deficit at ruin, and related quantities of interest.     

Analysis of risk models using a level crossing technique

This paper analyzes ruin-like risk models in Insurance, which are variants of the Cramer–Lundberg (C–L) model with a barrier or a threshold. We consider three model variants, which have different portfolio strategies when the risk reserve reaches the barrier or exceeds the threshold. In these models we construct a time-extended risk process defined on cycles of a specific renewal process. The time until ruin is equal to one cycle of the specific renewal process. We also consider a fourth model, which is a variant of a model proposed by Dickson and Waters (2004). The analysis of each model employs a level crossing method (LC) to derive the steady-state probability distribution of the time-extended risk process. From the derived distribution we compute the expected time until ruin, the probability distribution of the deficit at ruin, and related quantities of interest.     

On the discounted penalty function in the renewal risk model with general interclaim times

The defective renewal equation satisfied by the Gerber-Shiu discounted penalty function in the renewal risk model with arbitrary interclaim times is analyzed. The ladder height distribution is shown to be a mixture of residual lifetime claim severity distributions, which results in an invariance property satisfied by a large class of claim amount models. The class of exponential claim size distributions is considered, and the Laplace transform of the (discounted) defective density of the surplus immediately prior to ruin is obtained. The mixed Erlang claim size class is also examined. The simplified defective renewal equation which results when the penalty function only involves the deficit is used to obtain moments of the discounted deficit.     

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.     

常利率下复合泊松-更新风险模型的破产问题

本文研究了常数利率下,保费收入为复合Poisson过程,理赔到达过程为一般更新过程的风险模型.利用离散化的方法,获得了该风险模型的破产概率、破产时余额分布及破产前瞬间余额分布的级数展开式,推广了文[1]和文[2]中的相关结果.     

随机收益环境下的破产时刻罚金折现期望

This paper considers the expected discounted penalty function Φ(u) for the perturbed compound Poisson risk model with stochastic return on investments. After presenting an integro-differential equation that the expected discounted penalty function satisfies, the paper derives the closed form solution by constructing an identical equation. The exact expression for Φ (0) is given using the Laplace transform technique when interest rate is constant. Applications of the results are given to the ruin probability and moments of the deficit at ruin.     

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.     

常利率下复合泊松-更新风险模型的破产问题

本文研究了常数利率下, 保费收入为复合Poisson 过程, 理赔到达过程为一般更新过程的风险模型. 利用离散化的方法, 获得了该风险模型的破产概率、破产时余额分布及破产前瞬间余额分布的级数展开式, 推广了文[1] 和文[2] 中的相关结果.