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.     

一类索赔相关且带干扰的风险模型

该文讨论了索赔时间间隔与索赔量相关且带干扰的风险模型. 借助拉普拉斯变换研究了此模型的破产时刻、破产前瞬间盈余及破产时赤字三者的联合分布,得到了此联合分布拉普拉斯变换的一个分析表达式并讨论了当初始盈余值趋于无穷大时,此联合分布的渐近表达式.     

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.     

支付红利的复合二项模型

本文在完全离散的复合二项经典风险模型的基础上,考虑随机地支付红利的模型,当盈余大于或等于一个给定的非负整数红利界,并且没有索赔发生时,保险公司就以概率q0支付一个单位的红利,本文获得了这个模型的破产概率、破产时赤字的分布等的递推公式.     

离散三项分布风险模型中的贴现罚函数及其渐近解

Gerber和Shiu在1998年首次定义贴现罚函数为:m(u)=E{v~Tw(U_T-,|U_T|) I(T<∞)|U_0=u},其中w为一有界函数.通过对w和v的不同选择,可以得到一些与破产有关的变量的性质.本文用该方法对离散三项分布风险模型中的贴现罚函数问题进行了研究.主要得到了该模型中f(x,y;u)(即初始盈余为u,破产前瞬间盈余为x,破产时赤字为y这一事件的贴现概率)的明确表达式和该表达式的渐近解.还得到了导致破产发生的最后一个索赔额的分布.     

一类随机保费率下的风险模型

引入随机变量保费率,对古典风险模型进行推广,主要研究随机保费率下的风险模型,用随机过程和鞅论的方法得出破产概率、末离前最大盈余分布、破产前瞬时盈余与破产赤字的联合分布等精算量分布的具体表达式.     

稀疏过程的三特征的联合分布函数

本文考虑一类人寿保险,保费到达为Po isson过程,索赔到达为p-稀疏过程,我们推导三特征的联合分布函数;破产时间,破产概率,破产前的盈余,破产赤字,并由这联合分布得破产概率的显示表达式.     

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.     

完全离散的经典风险模型

本文系统地探讨了完全离散的经典风险模型，特别是重点研究了与风险有关的最终破产概率，破产前一刻的盈余和破产时赤字的概率律．Gerber仅在初始盈余为零的情况下给出了上述概率律的显式解，本文则对任意的初始盈余u≥0，给出了上述概率或概率律的递推解、变换解与显式解．     

离散的三项分布风险模型的渐近解

本文研究了离散的三项分布风险模型,在调节系数存在的前提下,借助于离散更新方程的一个极限定理,对于充分大的初始盈余给出了最终破产概率、破产前一刻的盈余和破产时赤字的概率的渐近解．其结果可以在离散的多项分布风险模型中得到推广．     

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．     

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.     

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.     

On the joint distribution of surplus before and after ruin under a Markovian regime switching model

We consider a Markovian regime switching insurance risk model (also called Markov-modulated risk model). The closed form solutions for the joint distribution of surplus before and after ruin when the initial surplus is zero or when the claim size distributions are phase-type distributed are obtained.     

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.     

Finite time ruin problems for the Erlang(2) risk model

We consider the Erlang(2) risk model and derive expressions for the density of the time to ruin and the joint density of the time to ruin and the deficit at ruin when the individual claim amount distribution is (i) an exponential distribution and (ii) an Erlang(2) distribution. We also consider the special case when the initial surplus is zero.     

Ruin problems for a discrete time risk model with random interest rate

In this paper, we study a discrete time risk model with random interest rate. The convergence of the discounted surplus process is proved by using martingale techniques, an expression of ruin probability is obtained, and bounds for ruin probability are included. In the second part of the paper, the distribution of surplus immediately after ruin, the distribution of surplus just before ruin, the joint distribution of the surplus immediately before and after ruin, and the distribution of ruin time are discussed.     

A Joint Density Function in the Renewal Risk Model

In this paper, we consider a general expression for $ϕ(u, x, y)$, the joint density function of the surplus prior to ruin and the deficit at ruin when the initial surplus is $u$. In the renewal risk model, this density function is expressed in terms of the corresponding density function when the initial surplus is 0. In the compound Poisson risk process with phase-type claim size, we derive an explicit expression for $ϕ(u, x, y)$. Finally, we give a numerical example to illustrate the application of these results.     

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.