The joint distributions of several important actuarial diagnostics in the classical risk model

In this paper we examine the joint distributions of several actuarial diagnostics which are important to insurers’ running in the classical risk model. They include the time of the surplus process leaving zero ultimately (simply, the ultimately leaving-time), the number of zero, the surplus immediately prior to ruin, the deficit at ruin, the supreme and minimum profits before ruin, the supreme profits and deficit until it leaves zero ultimately and so on. We obtain explicit expressions for their joint distributions mainly by strong Markov property of the surplus process—a technique used by Wu et al. (2002) [J. Appl. Math., in press], which is completely different from former contributions on this topic. Further, we give the exact calculating results for them when the individual claim amounts are exponentially distributed.     

与两种破产类型相关的余额极值联合分布

带干扰古典风险模型具有由索赔和小余额快速变化分别引起的两种破产和相应的破产时间．该文在两种类型破产各自发生的条件下,使用破产概率函数分别就余额过程首次返回零点以及最后一次返回零点所经历的时间间隔,给出了各自的余额最大值和最小值的联合分布．文章还给出了与该风险模型关联密切的若干鞅的表达式．     

利率相依的离散时间保险风险模型的破产问题

本文对利率具有一阶自回归的离散时间风险模型进行了研究,得到了破产前最大盈余的分布,破产前盈余、破产后赤字与破产前最大盈余的联合分布以及首达某一水平x的时间分布的递推公式.     

随机利率离散时间风险模型的破产问题

本文研究了引入随机利率的离散时间风险模型, 得到了破产持续时间的分布、盈余回复为正后的瞬间的盈余分布、 破产前最大盈余的分布、破产前盈余破产后赤字与破产前最大盈余的联合分布、 有限时间内穿出水平$x$的分布所满足的积分方程, 并同时证明了所得积分方程解的存在唯一性.     

连续时间复合二项模型多维精算量的联合分布

连续时间复合二项模型是由文献首先提出的.作为离散时间复合二项模型的连续化版本,连续时间复合二项模型的极限形式即为经典风险模型.为了得到该模型多维精算量的联合分布,该文引入了一列上穿零点,推导出该列上穿零点所构成的缺陷(defective)更新序列的更新质量函数.利用此更新质量函数及余额过程的强马氏性可以得到破产概率和包含破产时间,破产前余额,破产严重程度,破产前最大盈余,破产到恢复的最大赤字,整个过程的最大赤字等多维精算量的联合分布.由此联合分布得到其1-骨架链—离散时间复合二项模型的对应的联合分布,最后给出在1-骨架链中索赔额服从指数分布时这一特殊情况下相应多维精算量的联合分布的明确表达式.     

离散时间模型下最大赤字问题

本文对引入利率的离散时间风险模型得到了破产前最大盈余的分布 ,破产前盈余、破产后赤字与破产前最大盈余的联合分布以及首达某一水平 x的时间分布的递推公式 ,对不带利率的模型得到了最大赤字、发生最大赤字的时间的分布     

Ruin probability and joint distributions of some actuarial random vectors in the compound pascal model

The compound negative binomial model,introduced in this paper,is a discrete time version.We discuss the Markov properties of the surplus process,and study the ruin probability and the joint distributions of actuarial random vectors in this model.By the strong Markov property and the mass function of a defective renewal sequence,we obtain the explicit expressions of the ruin probability,the finite-horizon ruin probability,the joint distributions of T,U(T-1),|U(T)| and 0≤inn相似文献   

关于古典风险模型的一个联合分布

本文主要讨论古典风险模型矿产瞬间前的余额，破产时的赤字，破产前的最大余额，最后一次破产前的最大余额等的联合分布。     

带有随机干扰和确定投资回报风险过程下的一些分布

我们考虑既带有随机干扰又带有确定投资回报的风险过程, 得到了破产前瞬间盈余的分布$F_{\delta}(u,x)$及破产前瞬间盈余和破产时赤字的联合分布$H_{\delta}(u,x,y)$所满足的积分表达, 连续性及二次连续可微性和积分--微分方程. 同时, 只有随机干扰的风险模型下的破产前瞬间盈余的分布及破产前瞬间盈余和破产时赤字的联合分布所满足的性质也被得到. 已有文献中的诸多有关结果均可以通过令我们结论中的某些参数特殊化为零而得到.     

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.     

理赔为一般到达的常利率风险模型

本文研究了一般到达的常利率保险风险问题，应用建立Markov骨架过程的方法建立了理赔为一般到达的常利率风险模型．给出了破产时的余额分布、破产前瞬间的余额分布、破产时间与破产前瞬间余额的联合分布、破产时间与破产时余额的联合分布及破产前瞬间余额、破产时余额与破产时间的联合分布．     

Some Results for Classical Risk Process with Stochastic Return on Investments

In this paper, we discuss the classical risk process with stochastic return on investment. We prove some properties of the ruin probability, the supremum distribution before ruin and the surplus distribution at the time of ruin and derive the integro-differential equations satisfied by these distributions respectively.     

On the expected discounted penalty function at ruin of a surplus process with interest

In this paper, we study the expected value of a discounted penalty function at ruin of the classical surplus process modified by the inclusion of interest on the surplus. The ‘penalty’ is simply a function of the surplus immediately prior to ruin and the deficit at ruin. An integral equation for the expected value is derived, while the exact solution is given when the initial surplus is zero. Dickson’s [Insurance: Mathematics and Economics 11 (1992) 191] formulae for the distribution of the surplus immediately prior to ruin in the classical surplus process are generalised to our modified surplus process.     

The Maximum Surplus Distribution before Ruin in an Erlang（n） Risk Process Perturbed by Diffusion

In this paper, we consider the distribution of the maximum surplus before ruin in a generalized Erlang(n) risk process (i.e., convolution of n exponential distributions with possibly different parameters) perturbed by diffusion. It is shown that the maximum surplus distribution before ruin satisfies the integro-differential equation with certain boundary conditions. Explicit expressions are obtained when claims amounts are rationally distributed. Finally, the surplus distribution at the time of ruin and the surplus distribution immediately before ruin are presented.     

The Compound Poisson Surplus Model with Interest and Liquid Reserves: Analysis of the Gerber–Shiu Discounted Penalty Function

We modify the compound Poisson surplus model for an insurer by including liquid reserves and interest on the surplus. When the surplus of an insurer is below a fixed level, the surplus is kept as liquid reserves, which do not earn interest. When the surplus attains the level, the excess of the surplus over the level will receive interest at a constant rate. If the level goes to infinity, the modified model is reduced to the classical compound Poisson risk model. If the level is set to zero, the modified model becomes the compound Poisson risk model with interest. We study ruin probability and other quantities related to ruin in the modified compound Poisson surplus model by the Gerber–Shiu function and discuss the impact of interest and liquid reserves on the ruin probability, the deficit at ruin, and other ruin quantities. First, we derive a system of integro-differential equations for the Gerber–Shiu function. By solving the system of equations, we obtain the general solution for the Gerber–Shiu function. Then, we give the exact solutions for the Gerber–Shiu function when the initial surplus is equal to the liquid reserve level or equal to zero. These solutions are the key to the exact solution for the Gerber–Shiu function in general cases. As applications, we derive the exact solution for the zero discounted Gerber–Shiu function when claim sizes are exponentially distributed and the exact solution for the ruin probability when claim sizes have Erlang(2) distributions. Finally, we use numerical examples to illustrate the impact of interest and liquid reserves on the ruin probability.      

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.     

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.     

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.     

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.     

On a Dual Risk Model Perturbed by Diffusion with Dividend Threshold

In the dual risk model, the surplus process of a company is a L′evy process with sample paths that are skip-free downwards. In this paper, the authors assume that the surplus process is the sum of a compound Poisson process and an independent Wiener process. The dual of the jump-diffusion risk model under a threshold dividend strategy is discussed. The authors derive a set of two integro-differential equations satisfied by the expected total discounted dividend until ruin. The cases where profits follow an exponential or mixtures of exponential distributions are solved. Applying the key method of the Laplace transform, the authors show how the integro-differential equations are solved. The authors also discuss the conditions for optimality and show how an optimal dividend threshold can be calculated as well.