On a risk model with stochastic premiums income and dependence between income and loss

Labbé and Sendova (2009) [9] consider a compound Poisson risk model with stochastic premiums income. In this paper, we extend their model by assuming that there exists a specific dependence structure among the claim sizes, interclaim times and premium sizes. Assume that the distributions of the premium sizes and interclaim times are controlled by the claim sizes. When the individual premium sizes are exponentially distributed, the Laplace transforms and defective renewal equations for the (Gerber-Shiu) discounted penalty functions are obtained. When the individual premium sizes have rational Laplace transforms, we show that the Laplace transforms for the discounted penalty functions can also be obtained.     

The Gerber-Shiu penalty functions for two classes of renewal risk processes

In this paper, we study the Gerber-Shiu functions for a risk model with two independent classes of risks. We suppose that both of the two claim number processes are renewal processes with phase-type inter-claim times. By re-composing and analyzing the Markov chains associated with two given phase-type distributions, we obtain systems of integro-differential equations for two types of Gerber-Shiu functions. Explicit expressions for the Laplace transforms of the two types of Gerber-Shiu functions are established, respectively. And explicit results for the Gerber-Shiu functions are derived when the initial surplus is zero and when the two claim amount distributions are both from the rational family. Finally, an example is considered to illustrate the applicability of our main results.     

On the expected discounted penalty functions for two classes of risk processes under a threshold dividend strategy

In this paper, the discounted penalty (Gerber-Shiu) functions for a risk model involving two independent classes of insurance risks under a threshold dividend strategy are developed. We also assume that the two claim number processes are independent Poisson and generalized Erlang (2) processes, respectively. When the surplus is above this threshold level, dividends are paid at a constant rate that does not exceed the premium rate. Two systems of integro-differential equations for discounted penalty functions are derived, based on whether the surplus is above this threshold level. Laplace transformations of the discounted penalty functions when the surplus is below the threshold level are obtained. And we also derive a system of renewal equations satisfied by the discounted penalty function with initial surplus above the threshold strategy via the Dickson-Hipp operator. Finally, analytical solutions of the two systems of integro-differential equations are presented.     

Mathematical investigation of the Gerber-Shiu function in the case of dependent inter-claim time and claim size

In this paper we investigate the well-known Gerber-Shiu expected discounted penalty function in the case of dependence between the inter-claim times and the claim amounts. We set up an integral equation for it and we prove the existence and uniqueness of its solution in the set of bounded functions. We show that if δ>0, the limit property of the solution is not a regularity condition, but the characteristic of the solution even in the case when the net profit condition is not fulfilled. It is the consequence of the choice of the penalty function for a given density function. We present an example when the Gerber-Shiu function is not bounded, consequently, it does not tend to zero. Using an operator technique we also prove exponential boundedness.     

On a discrete risk model with two-sided jumps

In this paper, we consider a discrete renewal risk model with phase-type interarrival times and two-sided jumps. In this model, downward jumps represent claim loss, while upward jumps are also allowed to represent random gains. Assume that the downward jumps have an arbitrary probability function and the upward jumps have a rational probability generating function. We study the (Gerber-Shiu) discounted penalty function. The generating function, the recursive formula as well as an explicit expression for the discounted penalty function are obtained.     

Asymptotic aspects of the Gerber-Shiu function in the renewal risk model using Wiener-Hopf factorization and convolution equivalence

We study the asymptotic behavior of the Gerber-Shiu expected discounted penalty function in the renewal risk model. Under the assumption that the claim-size distribution has a convolution-equivalent density function, which allows both heavy-tailed and light-tailed cases, we establish some asymptotic formulas for the Gerber-Shiu function with a fairly general penalty function. These formulas become completely transparent in the compound Poisson risk model or for certain choices of the penalty function in the renewal risk model. A by-product of this work is an extension of the Wiener-Hopf factorization to include the times of ascending and descending ladders in the continuous-time renewal risk model.     

A Perturbed Risk Model with Dependence Based on a Generalized Farlie-Gumbel-Morgenstern Copula

In this paper, we consider a perturbed compound Poisson risk model with dependence, where the dependence structure for the claim size and the inter-claim time is modeled by a generalized Farlie-Gumbel-Morgenstern copula. The integro equations, the Laplace transforms and the defective renewal equations for the Gerber-Shiu functions are obtained. For exponential claims, some explicit expressions are obtained, and some numerical examples for the ruin probabilities are also provided.     

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??In this paper, we consider a perturbed compound Poisson risk model with dependence, where the dependence structure for the claim size and the inter-claim time is modeled by a generalized Farlie-Gumbel-Morgenstern copula. The integro equations, the Laplace transforms and the defective renewal equations for the Gerber-Shiu functions are obtained. For exponential claims, some explicit expressions are obtained, and some numerical examples for the ruin probabilities are also provided.     

The expected discounted penalty function for the perturbed compound Poisson risk process with constant interest

In this paper, we consider the Gerber-Shiu expected discounted penalty function for the perturbed compound Poisson risk process with constant force of interest. We decompose the Gerber-Shiu function into two parts: the expected discounted penalty at ruin that is caused by a claim and the expected discounted penalty at ruin due to oscillation. We derive the integral equations and the integro-differential equations for them. By solving the integro-differential equations we get some closed form expressions for the expected discounted penalty functions under certain assumptions.     

Threshold分红策略下带干扰的两类索赔风险模型的Geber-Shiu函数

本文研究了在threshold分红策略下带干扰的两类索赔风险模型的Geber-Shiu函数.这里假设两个索赔计数过程为独立的更新过程,其中一个为Poisson过程另一个为时间间隔服从广义Erlang(2)分布的更新过程.本文得到了threshold分红策略下Gerber-Shiu函数所满足的积分-微分方程及其边界条件....     

The perturbed compound Poisson risk model with constant interest and a threshold dividend strategy

In this paper, we consider the compound Poisson risk model perturbed by diffusion with constant interest and a threshold dividend strategy. Integro-differential equations with certain boundary conditions for the moment-generation function and the nth moment of the present value of all dividends until ruin are derived. We also derive integro-differential equations with boundary conditions for the Gerber-Shiu functions. The special case that the claim size distribution is exponential is considered in some detail.     

The Gerber-Shiu discounted penalty function in the risk process with phase-type interclaim times

In this paper, we consider the Gerber-Shiu discounted penalty function for the Sparre Anderson risk process in which the interclaim times have a phase-type distribution. By the Markov property of a joint process composed of the risk process and the underlying Markov process, we provide a new approach to prove the systems of integro-differential equations for the Gerber-Shiu functions. Closed form expressions for the Gerber-Shiu functions are obtained when the claim amount distribution is from the rational family. Finally we compute several numerical examples intended to illustrate the main results.     

具有常数红利边界的两类索赔相关风险模型数

研究常数红利边界下两类索赔相关的风险模型,两类索赔计数过程分别为独立的Poisson过程和广义Erlang(2)过程.利用分解Gerber-Shiu函数的方法,得到了Gerber-Shiu函数满足的积分-微分方程、边界条件、解析表达式及两类索赔额均服从指数分布时的破产概率表达式.     

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In this paper, an Erlang(2) risk model with time-dependent claims is studied under a multi-layer dividend strategy. First, some piecewise integro-differential equations with certain boundary conditions for the Gerber-Shiu function are derived. Then, applying these results, some defective renewal equations and explicit expressions for the Gerber-Shiu function are obtained when the joint density of the inter-claim time and claim size belongs to the rational family.     

Discrete Risk Model Revisited

In this paper, the fully discrete risk model is considered. Claim sizes are assumed to be integer-valued. A new method is employed to derive some explicit formulas of the Gerber-Shiu penalty function. Characteristic equations corresponding to recursive equations satisfied by Gerber-Shiu penalty function are analyzed and explicit expressions of the penalty function are then obtained. As a special case, the probability of ruin is obtained. National Natural Science Foundation of China(10571092,10271062)     

Further results on the euler and Genocchi numbers

Summary We characterize the ordinary generating functions of the Genocchi and median Genocchi numbers as unique solutions of some functional equations and give a direct algebraic proof of several continued fraction expansions for these functions. New relations between these numbers are also obtained.     

The perturbed compound Poisson risk model with linear dividend barrier

In this paper, we consider a diffusion perturbed classical compound Poisson risk model in the presence of a linear dividend barrier. Partial integro-differential equations for the moment generating function and the nth moment of the present value of all dividends until ruin are derived. Moreover, explicit solutions for the nth moment of the present value of dividend payments are obtained when the individual claim size distribution is exponential. We also provided some numerical examples to illustrate the applications of the explicit solutions. Finally we derive partial integro-differential equations with boundary conditions for the Gerber-Shiu function.     

复合马尔可夫二项模型的Gerber-Shiu折现罚金函数

本文研究复合马尔可夫二项模型的Gerber-Shiu折现罚金函数,得到了有条件和无条件的Gerber-Shiu折现罚金函数所满足的瑕疵更新方程.然后给出这些折现罚金函数的渐近表达式.     

On the DFR property of the compound geometric distribution with applications in risk theory

In 1988, Shanthikumar proved that the sum of a geometrically distributed number of i.i.d. DFR random variables is also DFR. In this paper, motivated by the inverse problem, we study monotonicity properties related to defective renewal equations, and obtain that if a compound geometric distribution is DFR, then the random variables of the sums are NWU (a class that contains DFR). Furthermore, we investigate some applications of risk theory and give a characterization of the exponential distribution.     

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.