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1.
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
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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. 相似文献
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In this article, the joint distributions of several actuarial diagnostics which are important to insurers’ running for the jump-diffusion risk process are examined. They include the ruin time, the time of the surplus process leaving zero ultimately (simply, the ultimately leaving-time), the surplus immediately prior to ruin, the supreme profits before ruin, the supreme profits and deficit until it leaves zero ultimately and so on. The explicit expressions for their distributions are obtained mainly by the various properties of L′evy process, such as the homogeneous strong Markov property and the spatial homogeneity property etc, moveover, the many properties for Brownian motion. 相似文献
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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. 相似文献
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本文研究保费到达为平衡更新过程的复合更新风险模型 ,给出了有限时间内的生存概率分布 ,破产时间 T与破产时资产盈余 U(T)的联合分布 ,及破产时间 T与破产前瞬时盈余 U(T- )的联合分布 . 相似文献
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稀疏过程的三特征的联合分布函数 总被引:1,自引:0,他引:1
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. 相似文献
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Jun Cai Runhuan Feng Gordon E. Willmot 《Methodology and Computing in Applied Probability》2009,11(3):401-423
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.
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Huai Xu & Ling Tang 《数学研究通讯:英文版》2013,29(1):88-96
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. 相似文献
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In this paper, we study absolute ruin questions for the perturbed compound Poisson risk process with investment and debit
interests by the expected discounted penalty function at absolute ruin, which provides a unified means of studying the joint
distribution of the absolute ruin time, the surplus immediately prior to absolute ruin time and the deficit at absolute ruin
time. We first consider the stochastic Dirichlet problem and from which we derive a system of integro-differential equations
and the boundary conditions satisfied by the function. Second, we derive the integral equations and a defective renewal equation
under some special cases, then based on the defective renewal equation we give two asymptotic results for the expected discounted
penalty function when the initial surplus tends to infinity for the light-tailed claims and heavy-tailed claims, respectively.
Finally, we investigate some explicit solutions and numerical results when claim sizes are exponentially distributed. 相似文献
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关于常利率风险模型在破产前后余额的分布 总被引:2,自引:0,他引:2
本文对常利率风险模型运用拉普拉斯变换给出了破产前后余额通过破产概率函数表示的有限公式,以及破产概率的分析表达式,另外对于破产前后余额分布的密度与破产前余额密度之间关系简要说明。 相似文献
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《Insurance: Mathematics and Economics》1988,7(2):75-80
In the classical compound Poisson model of the collective theory of risk let ψ(u, y) denote the probability that ruin occurs and that the negative surplus at the time of ruin is less than − y. It is shown how this function, which also measures the severity of ruin, can be calculated if the claim amount distribution is a translation of a combination of exponential distributions. Furthermore, these results can be applied to a certain discrete time model. 相似文献
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In this paper, a risk model where claims arrive according to a Markovian arrival process (MAP) is considered. A generalization of the well-known Gerber-Shiu function is proposed by incorporating the maximum surplus level before ruin into the penalty function. For this wider class of penalty functions, we show that the generalized Gerber-Shiu function can be expressed in terms of the original Gerber-Shiu function (see e.g. [Gerber, Hans U., Shiu, Elias, S.W., 1998. On the time value of ruin. North American Actuarial Journal 2(1), 48-72]) and the Laplace transform of a first passage time which are both readily available. The generalized Gerber-Shiu function is also shown to be closely related to the original Gerber-Shiu function in the same MAP risk model subject to a dividend barrier strategy. The simplest case of a MAP risk model, namely the classical compound Poisson risk model, will be studied in more detail. In particular, the discounted joint density of the surplus prior to ruin, the deficit at ruin and the maximum surplus before ruin is obtained through analytic Laplace transform inversion of a specific generalized Gerber-Shiu function. Numerical illustrations are then examined. 相似文献
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Jie-hua XieWei Zou 《Journal of Computational and Applied Mathematics》2011,235(8):2392-2404
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. 相似文献
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David C.M. Dickson 《Insurance: Mathematics and Economics》2012,50(3):334-337
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. 相似文献
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利润最大化风险最小化是保险公司运营所追求的目标,破产概率为公司进行风险决策提供了依据。本文基于随机利率环境下,保费随公司盈余水平调整的双分红复合帕斯卡模型,研究了股份制保险公司的有限时间破产概率。我们证明了公司盈余过程的齐次马氏性,得到了有限时间破产概率的计算方法,最后给出了具体算例。 相似文献