马尔可夫环境下聚合索赔的前二阶矩

We consider the discounted aggregate claims when the insurance risks and financial risks are governed by a discrete-time Markovian environment.We assume that the claim sizes and the financial risks fluctuate over time according to the states of economy,which are interpreted as the states of Markovian environment.We will then determine a system of differential equations for the Laplace-Stieltjes transform of the distribution of discounted aggregate claims under mild assumption.Moreover,using the differentio-integral equation,we will also investigate the first two order moments of discounted aggregate claims in a Markovian environment.     

马氏环境下带干扰Cox风险模型的罚金折现期望函数

研究了马氏环境下带干扰的Cox风险模型.首先给出了罚金折现期望函数满足的积分方程,然后给出了破产概率,破产前瞬时盈余、破产赤字的分布及各阶矩所满足的积分方程.最后给出当索赔额服从指数分布且理赔强度为两状态时的破产概率的拉普拉斯变换.     

On the classical risk model with credit and debit interests under absolute ruin

In this paper, we consider the dividend payments in a compound Poisson risk model with credit and debit interests under absolute ruin. We first obtain the integro-differential equations satisfied by the moment generating function and moments of the discounted aggregate dividend payments. Secondly, applying these results, we get the explicit expressions of them for exponential claims. Then, we give the numerical analysis of the optimal dividend barrier and the expected discounted aggregate dividend payments which are influenced by the debit and credit interests. Finally, we find the integro-differential equations satisfied by the Laplace transform of absolute ruin time and give its explicit expressions when the claim sizes are exponentially distributed.     

Moments of discounted aggregate claim costs until ruin in a Sparre Andersen risk model with general interclaim times

In the context of a Sparre Andersen risk model with arbitrary interclaim time distribution, the moments of discounted aggregate claim costs until ruin are studied. Our analysis relies on a novel generalization of the so-called discounted density which further involves a moment-based component. More specifically, while the usual discounted density contains a discount factor with respect to the time of ruin, we propose to incorporate powers of the sum until ruin of the discounted (and possibly transformed) claims into the density. Probabilistic arguments are applied to derive defective renewal equations satisfied by the moments of discounted aggregate claim costs until ruin. Detailed examples concerning the discounted aggregate claims and the number of claims until ruin are studied upon assumption on the claim severities. Numerical illustrations are also given at the end.     

Dividend payments in the classical risk model under absolute ruin with debit interest

This paper attempts to study the dividend payments in a compound Poisson surplus process with debit interest. Dividends are paid to the shareholders according to a barrier strategy. An alternative assumption is that business can go on after ruin, as long as it is profitable. When the surplus is negative, a debit interest is applied. At first, we obtain the integro‐differential equations satisfied by the moment‐generating function and moments of the discounted dividend payments and we also prove the continuous property of them at zero. Then, applying these results, we get the explicit expressions of the moment‐generating function and moments of the discounted dividend payments for exponential claims. Furthermore, we discuss the optimal dividend barrier when the claim sizes have a common exponential distribution. Finally, we give the numerical examples for exponential claims and Erlang (2) claims. Copyright © 2008 John Wiley & Sons, Ltd.     

变保费率Cox风险模型的研究

研究了当保费率随理赔强度的变化而变化时C ox风险模型的折现罚金函数,利用后向差分法得到了折现罚金函数所满足的积分方程,进而得到了破产概率,破产前瞬时盈余、破产时赤字的各阶矩所满足的积分方程.最后给出当理赔额服从指数分布,理赔强度为两状态的马氏过程时破产概率的拉普拉斯变换,对一些具体数值计算出了破产概率的表达式.     

On the Markov-modulated insurance risk model with tax

In this paper, we consider the Markov-modulated insurance risk model with tax. We assume that the claim inter-arrivals, claim sizes and premium process are influenced by an external Markovian environment process. The considered tax rule, which is the same as the one considered by Albrecher and Hipp [Blätter DGVFM 28(1):13–28, 2007], is to pay a certain proportion of the premium income, whenever the insurer is in a profitable situation. A system of differential equations of the non-ruin probabilities, given the initial environment state, are established in terms of the ruin probabilities under the Markov-modulated insurance risk model without tax. Furthermore, given the initial state, the differential equations satisfied by the expected accumulated discounted tax until ruin are also derived. We also give the analytical expressions for them by iteration methods.     

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.     

On the Gerber-Shiu Function and Optimal Dividend Strategy for a Thinning Risk Model

In this paper, the risk model under constant dividend barrier strategy is studied, in which the premium income follows a compound Poisson process and the arrival of the claims is a p-thinning process of the premium arrival process. The integral equations with boundary conditions for the expected discounted aggregate dividend payments and the expected discounted penalty function until ruin are derived. In addition, the explicit expressions for the Laplace transform of the ruin time and the expected aggregate discounted dividend payments until ruin are given when the individual stochastic premium amount and claim amount are exponentially distributed. Finally, the optimal barrier is presented under the condition of maximizing the expectation of the difference between discounted aggregate dividends until ruin and the deficit at ruin.     

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.     

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.     

常利息力影响下跳扩散模型的分红问题

本文中用常值利率驱动下的经典跳扩散模型模拟保险公司的盈余过程,研究了该模型在带壁分红策略下的若干问题.首先得出破产前分红折现的高阶矩所满足的积分微分方程,并在指数分布的情况下借助合流超几何函数给出了方程的显式解.其次关于破产前聚合分红得到了一些令人满意的结果,这些结果甚至对一般的分布都成立,另外讨论了分红流的次数和额度.最后研究了指数分布时破产赤字折现期望问题.本文的部分结论深化了精算学中一些已有研究成果.     

A uniform asymptotic estimate for discounted aggregate claims with subexponential tails

In this paper we study the tail probability of discounted aggregate claims in a continuous-time renewal model. For the case that the common claim-size distribution is subexponential, we obtain an asymptotic formula, which holds uniformly for all time horizons within a finite interval. Then, with some additional mild assumptions on the distributions of the claim sizes and inter-arrival times, we further prove that this formula holds uniformly for all time horizons. In this way, we significantly extend a recent result of Tang [Tang, Q., 2007. Heavy tails of discounted aggregate claims in the continuous-time renewal model. J. Appl. Probab. 44 (2), 285–294].     

马氏相依风险模型红利折现的矩

该文讨论常数红利边界下的马氏相依模型的矩的问题. 首先, 推导出破产前全部红利的折现期望、红利折现的高阶矩所满足的积分-微分方程组及相应的边界条件. 然后, 通过构造特殊的初始条件, 利用Laplace变换, 在给定的一类索赔分布下, 得到上面方程组的显式解. 最后, 给出两状态下指数索赔的数值计算结果.     

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.     

Discounted Aggregate Claim Costs Until Ruin in the Discrete-Time Renewal Risk Model

In this paper, we focus on analyzing the relationship between the discounted aggregate claim costs until ruin and ruin-related quantities including the time of ruin. To facilitate the evaluation of quantities of our interest as an approximation to the ones in the continuous case, discrete-time renewal risk model with certain dependent structure between interclaim times and claim amounts is considered. Furthermore, to provide explicit expressions for various moment-based joint probabilities, a fairly general class of distributions, namely the discrete Coxian distribution, is used for the interclaim times. Also, we assume a combination of geometrics claim size with arbitrary interlciam time distribution to derive a nice expression for the Gerber-Shiu type function involving the discounted aggregate claims until ruin. Consequently, the results are applied to evaluate some interesting quantities including the covariance between the discounted aggregate claim costs until ruin and the discounted claim causing ruin given that ruin occurs.     

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.     

允许赤字的MAP风险过程首达时

保险公司作为负债经营的特殊企业,其偿付能力受到监管部门的约束,本文以公司负债经营为前提研究其各种首次时.考虑MAP风险过程,即存在一随机背景Markov过程,索赔到达与索赔大小同时受这一背景过程影响,索赔到达为Markov到达点过程(MAP),索赔大小对于不同的背景状态具有不同的分布.本文给出首达时满足的积分-微分方程,通过求解带边界条件的积分-微分方程,给出了盈余过程从初始盈余水平到达某一给定盈余水平的首达时的Laplace变换的矩阵表示式,并由此推得了盈余过程到达指定水平的若干首达事件概率.     

The compound Poisson process perturbed by a diffusion with a threshold dividend strategy

In this paper, we consider the compound Poisson process perturbed by a diffusion in the presence of the so‐called threshold dividend strategy. Within this framework, we prove the twice continuous differentiability of the expected discounted value of all dividends until ruin. We also derive integro‐differential equations for the expected discounted value of all dividends until ruin and obtain explicit expressions for the solution to the equations. Along the same line, we establish explicit expressions for the Laplace transform of the time of ruin and the Laplace transform of the aggregate dividends until ruin. In the case of exponential claims, some examples are provided. Copyright © 2008 John Wiley & Sons, Ltd.     

Actuarial comparisons for aggregate claims with randomly right-truncated claims

In this note, we consider an extension of the largest claims reinsurance treaty (LCR) with random upper thresholds for the claim sizes, that we call retention levels. The Laplace transform order for insurer’s aggregate claims is obtained assuming dependence among the random retention levels. Different results about the influence of dependence on the insurer total claim amount are also given including the connections with LCR and the case of combination with quota-share. Algebraic bounds for the insurer aggregate claims are obtained when a common fixed threshold is considered.