On the expected discounted penalty function for risk process with tax

In this paper we consider the generalized Cramér-Lundberg risk model including tax payments. We investigate how tax payments affect the behavior of a Cramér-Lundberg surplus process by defining an expected discounted penalty function at ruin. We derive an explicit expression for this function by solving a differential equation. Consequently, the explicit formulas for the discounted probability density function of the surplus immediately before ruin and the discounted joint probability density function of the surplus immediately before ruin and the deficit at ruin are obtained. We also give explicit expressions for the function for exponential claims.     

Absolute ruin in the compound Poisson model with credit and debit interests and liquid reserves

In this paper, we study the absolute ruin probability in the compound Poisson model with credit and debit interests and liquid reserves. At first, we derive a system of integro‐differential equations with certain boundary conditions for the Gerber–Shiu function. Then, applying these results, we obtain asymptotical formula of the absolute ruin probability for subexponentially claims. Furthermore, when the claims are exponentially distributed, we obtain the explicit expressions for the Gerber–Shiu function and the exact solution for the absolute ruin probability. Finally, we discuss the absolute ruin probability by using the Gerber–Shiu function when debit interest is varying. In the case of exponential individual claim, we give the explicit expressions for the Gerber–Shiu function. Copyright © 2012 John Wiley & Sons, Ltd.     

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.     

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.     

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 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.     

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.     

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 ruin probability of the renewal model with constant interest force and negatively dependent heavy-tailed claims

Recently, Tang [Tang, Q., 2005a. Asymptotic ruin probabilities of the renewal model with constant interest force and regular variation. Scand. Actuar. J. (1), 1–5] obtained a simple asymptotic formula for the ruin probability of the renewal risk model with constant interest force and regularly varying tailed claims. In this paper, we use a completely different approach to extend Tang’s result to the case in which the claims are pairwise negatively dependent and extended regularly varying tailed.     

Asymptotic finite-time ruin probability for a bidimensional renewal risk model with constant interest force and dependent subexponential claims

This paper considers a bidimensional renewal risk model with constant interest force and dependent subexponential claims. Under the assumption that the claim size vectors form a sequence of independent and identically distributed random vectors following a common bivariate Farlie–Gumbel–Morgenstern distribution, we derive for the finite-time ruin probability an explicit asymptotic formula.     

关于带常数利率与盈余相依型loss-carry-forward税收系统的Cramr-Lundberg风险模型(英文)

本文研究了带常数利率和盈余相依型loss-carry-forward税收系统的Cramr-Lundberg风险模型.利用无穷小分析方法及该过程具有的的强马氏性,得出了保险公司从开始运营到破产期间税收折现总额的数学期望表达式.作为例子,本文给出了指数分布索赔假定下该税收折现函数的具体表达式.     

离散的相依风险模型的破产问题

研究一类索赔时间相依的离散风险模型,模型中假设每次主索赔可能引起一次副索赔,而每次副索赔有可能延迟发生.通过引入辅助模型,运用概率论的分析方法得到了破产前瞬时盈余和破产时赤字联合分布的递推解,以及初始值为0时最终破产概率的明确表达式.最后结合保险实例进行了数值模拟.     

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In this paper, for a kind of risk models with heavy-tailed and delayed claims, we derive the asymptotics of the infinite-time ruin probability and the uniform asymptotics of the finite-time ruin probability. The numerical simulation results are also presented. The results of theoretical analysis and numerical simulation show that the influence of the delay for the claim payment is nearly negligible to the ruin probability when the initial capital and running-time are all large.     

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.     

Ruin probabilities in the risk process with random income

We extend the classical risk model to the case in which the premium income process, modelled as a Poisson process, is no longer a linear function. We derive an analog of the Beekman convolution formula for the ultimate ruin probability when the inter-claim times are exponentially distributed. A defective renewal equation satisfied by the ultimate ruin probability is then given. For the general inter-claim times with zero-truncated geometrically distributed claim sizes, the explicit expression for the ultimate ruin probability is derived.     

The distribution of tax payments in a Lévy insurance risk model with a surplus-dependent taxation structure

We study the distribution of tax payments in the model of Kyprianou and Zhou [Kyprianou, A.E., Zhou, X., 2009. General tax structures and the Lévy insurance risk model. J. Appl. Probab. (in press)], that is a Lévy insurance risk model with a surplus-dependent tax rate. More precisely, after a short discussion on the so-called tax identity, we derive a recursive formula for arbitrary moments of the discounted tax payments until ruin and we identify the distribution of the tax payments when there is no force of interest.     

Ruin probability of the renewal model with risky investment and large claims

The ruin probability of the renewal risk model with investment strategy for a capital market index is investigated in this paper. For claim sizes with common distribution of extended regular variation, we study the asymptotic behaviour of the ruin probability. As a corollary, we establish a simple asymptotic formula for the ruin probability for the case of Pareto-like claims. This work was supported by National Natural Science Foundation of China (Grant Nos. 10571167, 70501028), the Beijing Sustentation Fund for Elitist (Grant No. 20071D1600800421), the National Social Science Foundation of China (Grant No. 05&ZD008) and the Research Grant of Renmin University of China (Grant No. 08XNA001)     

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.     

具有两种副索赔的离散相依风险模型破产问题

考虑了一类离散相依的风险模型,该模型假设主索赔以一定的概率引起两种副索赔,而第一种副索赔有可能延迟发生.通过引入一个辅助模型,分别得出了该风险模型初始盈余为0时破产前盈余与破产时赤字的联合分布的表达式、初始盈余为"时破产前盈余和破产时赤字的联合分布的递推公式、初始盈余为0时的破产概率,以及初始盈余为"时的破产概率求解方...