马氏调制费率复合Poisson-Geometric风险模型的预警区问题

考虑了一类具有马氏调制费率的复合Poisson-Geometric过程风险模型,充分利用盈余过程的强马氏性,得到第一个预警区的一个条件矩母函数所满足的微积分方程,并进一步在两状态情形下,当理赔额的分布为指数分布时得到了第一个预警区的一个条件矩母函数的具体表达式以解释结果.需要特别指出的是,所研究模型的盈余过程不具有平稳增量性,只能充分运用盈余过程的强马氏性,研究了一类具有马氏调制费率的复合Poisson-Geometric过程风险模型的预警区问题,丰富了保险公司对预警区问题的研究,对保险公司考虑财务预警系统以及保险监管部门设计某些监管指标系统具有一定的参考指导价值.     

常利率复合Poisson风险模型中的预警区问题

该文讨论了带常利率复合Poisson风险模型中的预警区问题.在此,作者提出了一种新的方法,其有别于Gerber于1990年提出的鞅方法,通过这种新方法,最终得到了负盈余持续时间的矩母函数及各阶矩,进而在索赔指数情形给出了精确解析式,并利用计算得到的数值结果讨论了利率变化对预警区的影响.     

复合Poisson-Geometric风险模型的预警区问题

应用有别于传统鞅方法的方法,充分利用盈余过程的强马氏性,在一类复合Poisson-Geometric风险模型下讨论预警区问题,得到第一个预警区的一个条件矩母函数所满足的微积分方程,并在指数索赔情形下给出其精确解.     

一类推广的常利率复合Poisson-Geometric风险模型的预警区问题

在复合Poisson-Geometric风险模型的基础上,引入利率因素,并将保费收入由线性过程推广为复合Poisson过程,建立了一类推广的带常利率复合Poisson-Geometric风险模型,该模型描述现实的能力更强,更具有实际意义.然后,利用盈余过程的强马氏性推导出了首个预警区的条件矩母函数所满足的积分方程,并进一步在保费额和索赔额都服从指数分布的情形下得出了其解析解.     

随机利率作用下的经典风险模型的破产概率

本文讨论了在随机利率作用下经典风险模型的破产问题,给出了导致公司破产的索赔额的L ap lace变换所满足的微分方程,给出了破产概率二次连续可微性的条件,得到了导致公司破产的所满足的积分微分方程;破产时刻公司赤字的L ap lace变换所满足的积分-微分方程.作为特例,本文给出了当索赔为指数分布地导致破产索赔额的L ap lace变换和破产时刻赤字的L ap lace变换的微分方程.     

On the time value of absolute ruin with tax

Consider a compound Poisson surplus process of an insurer with debit interest and tax payments. When the portfolio is in a profitable situation, the insurer may pay a certain proportion of the premium income as tax payments. When the portfolio is below zero, the insurer could borrow money at a debit interest rate to continue his/her business. Meanwhile, the insurer will repay the debts from his/her premium income. The negative surplus may return to a positive level except that the surplus is below a certain critical level. In the latter case, we say that absolute ruin occurs. In this paper, we discuss absolute ruin quantities by defining an expected discounted penalty function at absolute ruin. First, a system of integro-differential equations satisfied by the expected discounted penalty function is derived. Second, closed-form expressions for the expected discounted total sum of tax payments until absolute ruin and the Laplace-Stieltjes transform (LST) of the total duration of negative surplus are obtained. Third, for exponential individual claims, closed-form expressions for the absolute ruin probability, the LST of the time to absolute ruin, the distribution function of the deficit at absolute ruin and the expected accumulated discounted tax are given. Fourth, for general individual claim distributions, when the initial surplus goes to infinity, we show that the ratio of the absolute ruin probability with tax to that without tax goes to a positive constant which is greater than one. Finally, we investigate the asymptotic behavior of the absolute ruin probability of a modified risk model where the interest rate on a positive surplus is involved.     

The maximum surplus in a finite‐time interval for a discrete‐time risk model with exchangeable,dependent claim occurrences

This paper investigates a discrete‐time risk model that involves exchangeable dependent loss generating claim occurrences and compound binomially distributed aggregate loss amounts. First, a general framework is presented to derive the distribution of a surplus sequence using the model. This framework is then applied to obtain the distribution of any function of a surplus sequence in a finite‐time interval. Specifically, the distribution of the maximum surplus is obtained under nonruin conditions. Based on this distribution, the computation of the minimum surplus distribution is given. Asset and risk management–oriented implications are discussed for the obtained distributions based on numerical evaluations. In addition, comparisons are made involving the corresponding results of the classical discrete‐time compound binomial risk model, for which claim occurrences are independent and identically distributed.     

On the Distribution of Duration of First Negative Surplus for a Discrete Time Risk Model with Random Interest Rate

In this paper, we examine further annuity-due risk model presented by Cai (Probability in the Engineering and Informational Sciences, 16(2002), 309-324). We consider the computation for the distribution of duration of first negative surplus and the algorithm is shown for calculating probability that ruin occurs and the duration of first negative surplus takes any nonnegative integers values. Numerical illustration for the main result is given.     

一类索赔时间间隔服从混合指数分布的更新风险过程

本文研究了一类特殊的更新风险过程,其索赔时间间隔服从混合指数分布.首先,建立保险公司在时刻t的资产盈余模型,然后在该模型的基础上,根据Gerber的积分微分方程法和Laplace变换计算该公司的生存概率和赤字分布,最后分析盈余过程能顺利达到某一水平而不发生破产的概率.     

The Gerber–Shiu discounted penalty functions for a risk model with two classes of claims

In this paper, we consider the ruin problems for a risk model involving two independent classes of insurance risks. We assume that the claim number processes are independent Poisson and generalized Erlang(n) processes, respectively. When the generalized Lundberg equation has distinct roots with positive real parts, both of the Gerber–Shiu discounted penalty functions with zero initial surplus and the Laplace transforms of the Gerber–Shiu discounted penalty functions are obtained. Finally, some explicit expressions for the Gerber–Shiu discounted penalty functions with positive initial surplus are given when the claim size distributions belong to the rational family.     

带干扰phaes-type风险模型中的最大盈余及相关分布

本文考虑了索赔时间间距为phase-type分布时带干扰更新风险模型中的破产前最大盈余、破产后赤字的分布,建立了相应的积分-微分方程.最后,讨论了当索赔时间间距为Erlang(2)分布且索赔量满足指数分布时的特殊情形.     

变保费率扰动风险模型的有限时间破产概率和大偏差

该文考虑变保费率的扰动风险模型, 其中索赔的分布是重尾的. 对这个风险模型, 给出了索赔剩余过程的精细大偏差; 同时, 还得到了它的有限时间破产概率的Cramer-Lundberg型极限结果.     

Gerber–Shiu discounted penalty function in a Sparre Andersen model with multi-layer dividend strategy

This paper studies a Sparre Andersen model in which the inter-claim times are generalized Erlang(n) distributed. We assume that the premium rate is a step function depending on the current surplus level. A piecewise integro-differential equation for the Gerber–Shiu discounted penalty function is derived and solved. Finally, to illustrate the solution procedure, explicit expression for the Laplace transform of the time to ruin is given when the inter-claim times are generalized Erlang(2) distributed and the claim amounts are exponentially distributed.     

基于交替更新过程的模糊随机破产模型

假设索赔额、盈余额和更新过程均是在模糊随机环境中,并且将索赔过程定义为在交替更新过程.当索赔额和时间间隔是服从不同的指数分布时,本文建立了交替更新过程下的模糊随机破产模型,并给出了最终破产概率公式与最终破产机会均值公式.     

Structural properties of Gerber-Shiu functions in dependent Sparre Andersen models

The structure of various Gerber-Shiu functions in Sparre Andersen models allowing for possible dependence between claim sizes and interclaim times is examined. The penalty function is assumed to depend on some or all of the surplus immediately prior to ruin, the deficit at ruin, the minimum surplus before ruin, and the surplus immediately after the second last claim before ruin. Defective joint and marginal distributions involving these quantities are derived. Many of the properties in the Sparre Andersen model without dependence are seen to hold in the present model as well. A discussion of Lundberg’s fundamental equation and the generalized adjustment coefficient is given, and the connection to a defective renewal equation is considered. The usual Sparre Andersen model without dependence is also discussed, and in particular the case with exponential claim sizes is considered.     

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.     

Exit Times,Overshoot and Undershoot for a Surplus Process in the Presence of an Upper Barrier

We study the movement of a surplus process with initial capital u in the presence of two barriers: a lower barrier at zero and an upper barrier at b (b > u). More specifically, we consider the behaviour of the surplus: (a) in continuous time; and (b) only at claim arrival times. For each of these cases, we find the expected time until the process exits the interval [0,b]. We also obtain results related to the undershoot and overshoot of the surplus which, in particular for case (b) above, are derived under the assumption that the distribution of claim sizes and/or claim interarrival times belongs to the mixed Erlang class. In the final section we discuss the implementation of the methods in a number of examples using computer algebra software. These examples illustrate the efficiency of the methods even in fairly complicated cases.     

Stochastic analysis of life insurance surplus

The aim of the paper is to examine the behavior of insurance surplus over time for a portfolio of homogeneous life policies. We distinguish between stochastic and accounting surpluses and derive their first two moments. A recursive formula is proposed for calculating the distribution function of the accounting surplus. We then examine the probability that the accounting surplus becomes negative in a given insurance year. Numerical examples illustrate the results for portfolios of temporary and endowment life policies assuming a conditional AR(1) process for the rates of return.     

Precise large deviations for generalized dependent compound renewal risk model with consistent variation

We investigate the precise large deviations of random sums of negatively dependent random variables with consistently varying tails. We find out the asymptotic behavior of precise large deviations of random sums is insensitive to the negative dependence. We also consider the generalized dependent compound renewal risk model with consistent variation, which including premium process and claim process, and obtain the asymptotic behavior of the tail probabilities of the claim surplus process.     

The Risk Model with Interest,Liquid Reserves and a Constant Dividend Barrier

In this paper, we consider the compound Poisson surplus model with interest, liquid reserves and a constant dividend barrier. When the surplus of an insurer is below a fixed level, the surplus is kept as liquid reserves, which does not earn interest. When the surplus attains the level, the surplus will receive interest at a constant rate. When the surplus hits another fixed higher lever, the excess of the surplus over this higher level will be distributed to the shareholders as dividends. We derive a system of integro-differential equations for the Gerber-Shiu discounted penalty function and obtain the solutions to these integro-differential equations. In the case where the claim sizes are exponential distributed, we get the exact solutions of zero discounted Gerber-Shiu function. We also get the integro-differential equation for the expectation of the discounted dividends until ruin which is the key to discuss the optimal dividend barrier. And we give the exact solution in the special case with exponential claim sizes.