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.     

具有随机消费的带恒定红利界的对偶干扰风险模型

考虑了具有随机消费的带恒定红利界的对偶干扰风险模型.分别建立了破产前红利支付与期望折现罚函数所满足的积分-微分方程.当消费量与收入量均为指数分布时,得到了破产前红利支付与破产时间的解析表达式,并列举了数值例子.     

带观察时的跳服从Erlang(n)分布的对偶模型的红利贴现问题

研究了跳服从Erlang(n)分布,随机观察时服从指数分布的对偶风险模型.假设在边值策略下红利分发只在观察时发生,建立了红利期望贴现函数V(u;b)的微积分方程组.给出了当收益额服从PH(m)分布时V(u;b)的解析解.探讨了当收益额服从指数分布时V(u;b)的具体求解方法.     

A Markov decision problem in a risk model with interest rate and Markovian environment

We consider the compound binomial model in a Markovian environment presented by Cossette et al.(2004). We modify the model via assuming that the company receives interest on the surplus and a positive real-valued premium per unit time, and introducing a control strategy of periodic dividend payments. A Markov decision problem arises and the control objective is to maximize the cumulative expected discounted dividends paid to the shareholders until ruin minus a discounted penalty for ruin. We show that under the absence of a ceiling of dividend rates the optimal strategy is a conditional band strategy given the current state of the environment process. Under the presence of a ceiling for dividend rates, the character of the optimal control strategy is given. In addition, we offer an algorithm for the optimal strategy and the optimal value function.Numerical results are provided to illustrate the algorithm and the impact of the penalty.     

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与经典Cramer-Lundberg风险模型中保费收取过程 是时间的线性函数不同, 我们考虑聚合的保费收取过程是复合Poisson过程, 研究了在此模型下的常数分红策略问题. Dickson和Waters,(2004)指出在破产发生时, 股东还应有责任偿付破产时的赤字. 因此, 在本文中考虑的最优准则是最大化破产发生前的分红折现值与破产发生时赤字的差的期望. 做为例子, 当个体保费收取额和索赔额均为指数分布时, 给出了计算分红障碍的条件     

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.     

Dividend maximization under consideration of the time value of ruin

In the Cramér-Lundberg model and its diffusion approximation, it is a classical problem to find the optimal dividend payment strategy that maximizes the expected value of the discounted dividend payments until ruin. One often raised disadvantage of this approach is the fact that such a strategy does not take the lifetime of the controlled process into account. In this paper we introduce a value function which considers both expected dividends and the time value of ruin. For both the diffusion model and the Cramér-Lundberg model with exponential claim sizes, the problem is solved and in either case the optimal strategy is identified, which for unbounded dividend intensity is a barrier strategy and for bounded dividend intensity is of threshold type.     

On the renewal risk model under a threshold strategy

In this paper, we consider the renewal risk process under a threshold dividend payment strategy. For this model, the expected discounted dividend payments and the Gerber–Shiu expected discounted penalty function are investigated. Integral equations, integro-differential equations and some closed form expressions for them are derived. When the claims are exponentially distributed, it is verified that the expected penalty of the deficit at ruin is proportional to the ruin probability.     

Statistical estimation for some dividend problems under the compound Poisson risk model

In this paper, we consider some dividend problems in the classical compound Poisson risk model under a constant barrier dividend strategy. Suppose that the Poisson intensity for the claim number process and the distribution for the individual claim sizes are both unknown. We use the COS method to study the statistical estimation for the expected present value of dividend payments before ruin and the expected discounted penalty function. The convergence rates under large sample setting are derived. Some simulation results are also given to show effectiveness of the estimators under finite sample setting.     

绝对破产下具有贷款利息及常数分红界的扰动复合Poisson风险模型

该文研究了绝对破产下具有贷款利息及常数分红界的扰动复合Poisson风险模型,得到了折现分红总量的均值函数,及其矩母函数以及此模型的期望折现罚金函数(Gerber-Shiu函数)满足的积分-微分方程及边值条件,并求出了某些特殊情形下的具体表达式.     

Dividend Problem with Parisian Delay for a Spectrally Negative Lévy Risk Process

In this paper, we consider dividend problem for an insurance company whose risk evolves as a spectrally negative Lévy process (in the absence of dividend payments) when a Parisian delay is applied. An objective function is given by the cumulative discounted dividends received until the moment of ruin, when a so-called barrier strategy is applied. Additionally, we consider two possibilities of a delay. In the first scenario, ruin happens when the surplus process stays below zero longer than a fixed amount of time. In the second case, there is a time lag between the decision of paying dividends and its implementation.     

Dividend optimization under the gamma-distribution of claims

The classic insurance company work model with gamma-distribution of claim amount is considered. It is supposed that the company applies a dividend barrier strategy. The form of the expected discounted dividends accumulated until the ruin and the expected discounted deficit at the ruin are found. We deal with the optimal barriers which maximize either the dividends amount or shareholders profit. The barrier optimization is illustrated by some examples.     

Dividend payments with a threshold strategy in the compound Poisson risk model perturbed by diffusion

In the absence of dividends, the surplus of an insurance company is modelled by a compound Poisson process perturbed by diffusion. Dividends are paid at a constant rate whenever the modified surplus is above the threshold, otherwise no dividends are paid. Two integro-differential equations for the expected discounted dividend payments prior to ruin are derived and closed-form solutions are given. Accordingly, the Gerber–Shiu expected discounted penalty function and some ruin related functionals, the probability of ultimate ruin, the time of ruin and the surplus before ruin and the deficit at ruin, are considered and their analytic expressions are given by general solution formulas. Finally the moment-generating function of the total discounted dividends until ruin is discussed.     

Dividend problems in the dual model with diffusion and exponentially distributed observation time

Consider dividend problems in the dual model with diffusion and exponentially distributed observation time where dividends are paid according to a barrier strategy. Assume that dividends can only be paid with a certain probability at each point of time, that is, on each observation, if the surplus exceeds the barrier, the excess is paid as dividend. In this paper, integro-differential equations for the expected discounted sum of dividends paid until ruin and the Laplace transform of ruin time are derived. When the gains are exponentially distributed, explicit expressions for the ruin probability, the expected discounted sum of dividends paid until ruin, the Laplace transform of ruin time and the expectation of ruin time are also obtained.     

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.     

带常数边界的平衡更新风险模型的破产问题

本文讨论带常数边界的平衡更新风险模型的破产问题.利用Markov性质,给出惩罚函数满足的积分-微分方程,证明其惩罚函数可由更新风险模型的惩罚函数表示,并且给出一个具体的例子.     

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.     

On a risk model with Markovian arrivals and tax

A risk model with Markovian arrivals and tax payments is considered.When the insurer is in a profitable situation,the insurer may pay a certain proportion of the premium income as tax payments. First,t...     

一类混合分红策略下的广义Erlang(n)风险模型

本文考虑混合分红策略下索赔来到间隔为广义Erlang(n)分布的更新风险模型,利用指数分布的无记忆性,分别得到破产前期望折现分红函数和折现分红的矩母函数满足的积分-微分方程及其边界条件.最后给出索赔为指数分布及索赔来到间隔为广义Erlang(2)分布的风险模型的期望折现分红函数的精确表达式.     

On a dual model with a dividend threshold

In insurance mathematics, a compound Poisson model is often used to describe the aggregate claims of the surplus process. In this paper, we consider the dual of the compound Poisson model under a threshold dividend strategy. We derive a set of two integro-differential equations satisfied by the expected total discounted dividends until ruin and show how the equations can be solved by using only one of the two integro-differential equations. The cases where profits follow an exponential or a mixture of exponential distributions are then solved and the discussion for the case of a general profit distribution follows by the use of Laplace transforms. We illustrate how the optimal threshold level that maximizes the expected total discounted dividends until ruin can be obtained, and finally we generalize the results to the case where the surplus process is a more general skip-free downwards Lévy process.