A cyclic approach on classical ruin model

The ruin problem has long since received much attention in the literature. Under the classical compound Poisson risk model, elegant results have been obtained in the past few decades. We revisit the finite-time ruin probability by using the idea of cycle lemma, which was used in proving the ballot theorem. The finite-time result is then extended to infinite-time horizon by applying the weak law of large numbers. The cycle lemma also motivates us to study the claim instants retrospectively, and this idea can be used to reach the ladder height distribution on the infinite-time horizon. The new proofs in this paper link the classical finite-time and infinite-time ruin results, and give an intuitive way to understand the nature of ruin.     

Ruin probabilities in the discrete time renewal risk model

In this paper, we study the discrete time renewal risk model, an extension to Gerber’s compound binomial model. Under the framework of this extension, we study the aggregate claim amount process and both finite-time and infinite-time ruin probabilities. For completeness, we derive an upper bound and an asymptotic expression for the infinite-time ruin probabilities in this risk model. Also, we demonstrate that the proposed extension can be used to approximate the continuous time renewal risk model (also known as the Sparre Andersen risk model) as Gerber’s compound binomial model has been proposed as a discrete-time version of the classical compound Poisson risk model. This allows us to derive both numerical upper and lower bounds for the infinite-time ruin probabilities defined in the continuous time risk model from their equivalents under the discrete time renewal risk model. Finally, the numerical algorithm proposed to compute infinite-time ruin probabilities in the discrete time renewal risk model is also applied in some of its extensions.     

在风险投资下破产概率的一个渐近显式

在本文中, 我们研究了一个离散时间风险模型的破产概率\bd 在此风险模型中, 保险公司的剩余资本被用于进行风险投资\bd 我们运用纯概率的手法建立了无限时间破产概率的渐近显式, 从而将Tang和Tsitsiashvili (2003)近期的一个结果推广到了无限时间的场合.     

保险系统中一种推广风险模型的破产概率

将经典复合 Poisson风险模型推广至更为一般情况 ,其中保单以 Poisson分布流到达且收取的保费为随机变量 ,建立一种双复合 Poisson风险模型 .对此模型 ,得到了最终破产概率的一般表达式和破产概率的一个上界估计值 .     

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 partial integrodifferential equation of Seal’s type

In this paper we generalize a partial integrodifferential equation satisfied by the finite time ruin probability in the classical Poisson risk model. The generalization also includes the bivariate distribution function of the time of and the deficit at ruin. We solve the partial integrodifferential equation by Laplace transforms with the help of Lagrange’s implicit function theorem. The assumption of mixed Erlang claim sizes is then shown to result in tractable computational formulas for the finite time ruin probability as well as the bivariate distribution function of the time of and the deficit at ruin. A more general partial integrodifferential equation is then briefly considered.     

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 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.      

Ruin probability in the presence of interest earnings and tax payments

In this paper we investigate the ruin probability in a general risk model driven by a compound Poisson process. We derive a formula for the ruin probability from which the Albrecher–Hipp tax identity follows as a corollary. Then we study, as an important special case, the classical risk model with a constant force of interest and loss-carried-forward tax payments. For this case we derive an exact formula for the ruin probability when the claims are exponential and an explicit asymptotic formula when the claims are subexponential.     

离散时间的双Poisson模型的破产概率

本文在离散复合Poisson风险模型的基础上,研究保费的收取也为一个Poisson过程的模型, 在保费收取量和理赔量都离散取整数值时,我们运用转移概率推导出了保险公司在有限时间内破产的概率以及最终破产概率的级数表达式和矩阵表达式.     

Tail bounds for the joint distribution of the surplus prior to and at ruin

For the classical risk model with Poisson arrivals, we study the (bivariate) tail of the joint distribution of the surplus prior to and at ruin. We obtain some exact expressions and new bounds for this tail, and we suggest three numerical methods that may yield upper and lower bounds for it. As a by-product of the analysis, we obtain new upper and lower bounds for the probability and severity of ruin. Many of the bounds in the present paper improve and generalise corresponding bounds that have appeared earlier. For the numerical bounds, their performance is also compared against bounds available in the literature.     

带干扰的双复合Poisson风险模型

对古典风险模型进行推广,主要研究保费收入过程为带干扰双复合Poisson过程的风险模型,运用鞅的方法得出了破产概率满足的Lundburg不等式.     

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.     

Uniform asymptotics for the finite-time and infinite-time ruin probabilities in a dependent risk model with constant interest rate and heavy-tailed claims

We consider a nonstandard risk model with constant interest rate. For the case where the claim sizes follow a common heavy-tailed distribution and fulfill a dependence structure proposed by Geluk and Tang [J. Geluk and Q. Tang, Asymptotic tail probabilities of sums of dependent subexponential random variables, J. Theor. Probab., 22:871–882, 2009] while the interarrival times fulfill the so-called widely lower orthant dependence, we establish a weakly asymptotically equivalent formula for the infinite-time ruin probability. In particular, when the dependence structure for claim sizes is strengthened to the widely upper orthant dependence, this result implies a uniformly asymptotically equivalent formula for the finite-time and infinite-time ruin probabilities.     

带红利线的双复合Poisson过程风险模型的破产概率

在考虑红利付款下,将经典风险模型推广为双复合Po isson过程模型,应用鞅论的方法,得出了最终破产概率和Lundberg不等式.     

A note on the convexity of ruin probabilities

Conditions for the convexity of compound geometric tails and compound geometric convolution tails are established. The results are then applied to analyze the convexity of the ruin probability and the Laplace transform of the time to ruin in the classical compound Poisson risk model with and without diffusion. An application to an optimization problem is given.     

双复合Poisson风险模型

研究了保费收取过程是复合Po isson过程,索赔总额是复合Po isson过程的风险模型,给出了不破产概率的积分表示,以及在特殊情况下不破产概率的具体表达式,并用鞅方法得出了破产概率满足的Lundberg不等式和一般公式.     

多险种复合Poisson风险模型和破产概率

经典风险模型只描述了单一险种的经营模式,具有局限性,本文对多险种的复合Poisson风险模型的破产概率进行了研究。本文给出了初始资本为0时破产概率皿（O）的明确表达式,以及理赔量服从指数分布且初始资本为u时破产概率ψ（u）的明确表达式。     

Ruin probabilities of a bidimensional risk model with investment

We consider a classical risk model with the possibility of investment. We study two types of ruin in the bidimensional framework. Using the martingale technique, we obtain an upper bound for the infinite-time ruin probability with respect to the ruin time T_max(u₁,u₂). For each type of ruin, we derive an integral-differential equation for the survival probability, and an explicit asymptotic expression for the finite-time ruin probability.     

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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.