On the ruin probabilities of a bidimensional perturbed risk model

We follow some recent works to study the ruin probabilities of a bidimensional perturbed insurance risk model. For the case of light-tailed claims, using the martingale technique we obtain for the infinite-time ruin probability a Lundberg-type upper bound, which captures certain information of dependence between the two marginal surplus processes. For the case of heavy-tailed claims, we derive for the finite-time ruin probability an explicit asymptotic estimate.     

A two-dimensional ruin problem on the positive quadrant

In this paper we study the joint ruin problem for two insurance companies that divide between them both claims and premia in some specified proportions (modeling two branches of the same insurance company or an insurance and re-insurance company). Modeling the risk processes of the insurance companies by Cramér-Lundberg processes we obtain the Laplace transform in space of the probability that either of the insurance companies is ruined in finite time. Subsequently, for exponentially distributed claims, we derive an explicit analytical expression for this joint ruin probability by explicitly inverting this Laplace transform. We also provide a characterization of the Laplace transform of the joint ruin time.     

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．     

Ruin problems for an autoregressive risk model with dependent rates of interest

In this paper, we consider a discrete insurance risk model in which the claims, the premiums and the rates of interest are assumed to have dependent autoregressive structures (AR(1)). We derive recursive and integral equations for expected discounted penalty function. By these equations, we obtain generalized Lundberg inequality for the infinite time severity of ruin and hence for the infinite time ruin probability, consider asymptotic formula for the finite time ruin probability when loss distributions have regularly varying tails, and study some probability properties of the duration of ruin.     

Ruin probability in the presence of interest earnings and tax payments

In this paper we investigate the ruin probability in a general risk model driven by a compound Poisson process. We derive a formula for the ruin probability from which the Albrecher–Hipp tax identity follows as a corollary. Then we study, as an important special case, the classical risk model with a constant force of interest and loss-carried-forward tax payments. For this case we derive an exact formula for the ruin probability when the claims are exponential and an explicit asymptotic formula when the claims are subexponential.     

The joint distribution of the time to ruin and the number of claims until ruin in the classical risk model

We use probabilistic arguments to derive an expression for the joint density of the time to ruin and the number of claims until ruin in the classical risk model. From this we obtain a general expression for the probability function of the number of claims until ruin. We also consider the moments of the number of claims until ruin and illustrate our results in the case of exponentially distributed individual claims. Finally, we briefly discuss joint distributions involving the surplus prior to ruin and deficit at ruin.     

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??A new risk model is constructed, where the total number of claims satisfies the geometric first-order integer-valued autoregressive process. Moreover, we obtain the equation of the adjustment coefficient. We discuss the relationships among the dependence on the number of claims in each period, the adjustment coefficient, and ruin probability by numerical simulations. The results show that, with the increase of the dependence on the number of claims in each period, the adjustment coefficient decrease and ruin probability increase gradually.     

带扩散扰动项的广义双Poisson风险模型下的破产概率

本文首先在[1]-[4]讨论的基础上,将经典的破产模型推广到带扩散扰动项的广义双Po isson风险模型,即将保费收取过程和索赔总额过程同时推广到广义复合Po isson过程,以此解决在同一时刻有两张以上保单到达和两个以上顾客索赔的实际问题;接着运用鞅方法证明了破产概率满足的Lundberg不等式和一般公式在我们所建的模型下同样成立.     

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.     

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.     

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.     

Asymptotics for the ruin probabilities of a two‐dimensional renewal risk model with heavy‐tailed claims

In this paper, we consider two dependent classes of insurance business with heavy‐tailed claims. The dependence comes from the assumption that claim arrivals of the two classes are governed by a common renewal counting process. We study two types of ruin in the two‐dimensional framework. For each type of ruin, we establish an asymptotic formula for the finite‐time ruin probability. These formulae possess a certain uniformity feature in the time horizon. Copyright © 2010 John Wiley & Sons, Ltd.     

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.     

保费率-巨灾索赔相依的风险模型的破产概率

进一步推广Sparre Andersen风险模型,考虑有意外巨额赔付情况下得到保险公司的破产概率,并得到尾等价式,此结果反映了特殊的巨额索赔对破产的影响程度.另外,当有巨灾索赔发生的时候,模型会对保险费率做出相应的调整.     

A time-series risk model with constant interest for dependent classes of business

In this paper, we propose a discrete-time model with dependent classes of business using a time-series approach. Specifically, premiums and claims of all classes are supposed to satisfy a multivariate first-order autoregressive time-series model. A constant interest rate is also included in the model. A Lundberg-type inequality for the ruin probability is deduced. We also give an example with constant premiums and two classes of claims for which an expression as well as an exponential bound for the ruin probability is given. A simulation study is provided to help understanding the model.     

马氏利率下相关险种的离散风险建模和破产概率

在随机利率服从有限齐次Markov链下,建立相关险种离散风险模型,采用递推方法得到了有限时间破产概率的递推等式和最终破产概率的积分等式;给出了有限时间破产概率和最终破产概率的上界,导出了破产时刻余额分布的计算等式.     

On the decomposition of the absolute ruin probability in a perturbed compound Poisson surplus process with debit interest

We consider a compound Poisson surplus process perturbed by diffusion with debit interest. When the surplus is below zero or the company is on deficit, the company is allowed to borrow money at a debit interest rate to continue its business as long as its debt is at a reasonable level. When the surplus of a company is below a certain critical level, the company is no longer profitable, we say that absolute ruin occurs at this situation. In this risk model, absolute ruin may be caused by a claim or by oscillation. Thus, the absolute ruin probability in the model is decomposed as the sum of two absolute ruin probabilities, where one is the probability that absolute ruin is caused by a claim and the other is the probability that absolute ruin is caused by oscillation. In this paper, we first give the integro-differential equations satisfied by the absolute ruin probabilities and then derive the defective renewal equations for the absolute ruin probabilities. Using these defective renewal equations, we derive the asymptotical forms of the absolute ruin probabilities when the distributions of claim sizes are heavy-tailed and light-tailed. Finally, we derive explicit expressions for the absolute ruin probabilities when claim sizes are exponentially distributed.     

相关风险和模型的破产概率

本文研究了两险种理赔到达过程相关的正风险和模型与负风险和模型.利用Lundberg指数是相关因子的单调递减函数的性质,证明了破产概率是随着相关因子的增加而增大的,从而将相应的结果推广到了两险种理赔到达过程相关的风险和模型.     

含有正、负风险和风险过程的破产概率

本文考虑含正风险和与负风险和风险过程的破产问题, 给出该风险过程的破产概率所满足的积分--微分方程和指数不等式, 研究正风险和类与负风险和类之间的相关性对破产概率的影响, 并对具体实例给出数值比较结果.     

Distributions for the risk process with a stochastic return on investments

In this paper, we consider a risk model with stochastic return on investments. We mainly discuss the ruin probability, the surplus distribution at the time of ruin and the supremum distribution of the surplus before ruin. We prove some properties for these distributions and derive the integro-differential equations satisfied by them. We present the relation between the ruin probability and the supremum distribution before ruin.