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.     

Dividend payments with a threshold strategy in the compound Poisson risk model perturbed by diffusion

In the absence of dividends, the surplus of an insurance company is modelled by a compound Poisson process perturbed by diffusion. Dividends are paid at a constant rate whenever the modified surplus is above the threshold, otherwise no dividends are paid. Two integro-differential equations for the expected discounted dividend payments prior to ruin are derived and closed-form solutions are given. Accordingly, the Gerber–Shiu expected discounted penalty function and some ruin related functionals, the probability of ultimate ruin, the time of ruin and the surplus before ruin and the deficit at ruin, are considered and their analytic expressions are given by general solution formulas. Finally the moment-generating function of the total discounted dividends until ruin is discussed.     

复合Poisson模型中“双界限”分红问题

引入了复合Poisson模型中的"双界限"分红模型,在这种模型中,当盈余超过上限时分红以不超过保费率的速率付出,低于下限后保费率增大.文中利用Gerber- Shiu函数来分析这种模型,先导出了Gerber-Shiu函数m_1,m_2,m_3满足的积分-微分方程,再给出m_1,m_2,m_3的解析表示,最后通过几步把Gerber-Shiu函数m(u;b_1,b)的解析式表示出来.     

On two actuarial quantities for the compound Poisson risk model with tax and a threshold dividend strategy

In this paper, we consider a compound Poisson risk model with taxes paid according to a loss-carry-forward system and dividends paid under a threshold strategy. First, the closed-form expression of the probability function for the total number of taxation periods over the lifetime of the surplus process is derived. Second, analytical expression of the expected accumulated discounted dividends paid between two consecutive taxation periods is provided. In addition, explicit expressions are also given for the exponential individual claims.     

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.     

On the threshold dividend strategy for a generalized jump-diffusion risk model

In this paper, we generalize the Cramér-Lundberg risk model perturbed by diffusion to incorporate jumps due to surplus fluctuation and to relax the positive loading condition. Assuming that the surplus process has exponential upward and arbitrary downward jumps, we analyze the expected discounted penalty (EDP) function of Gerber and Shiu (1998) under the threshold dividend strategy. An integral equation for the EDP function is derived using the Wiener-Hopf factorization. As a result, an explicit analytical expression is obtained for the EDP function by solving the integral equation. Finally, phase-type downward jumps are considered and a matrix representation of the EDP function is presented.     

Ruin measures for a compound Poisson risk model with dependence based on the Spearman copula and the exponential claim sizes

This paper is devoted to an extension to the classical compound risk model. We relax the independence assumption of claim amounts and interclaim times. The dependent structure between these random variables is described by the Spearman copula. We study the Laplace transform of the discounted penalty function and we give the explicit expression of it for the exponential claim size.     

The compound Poisson risk model with dependence under a multi-layer dividend strategy

In this paper, a compound Poisson risk model with time-dependent claims is studied under a multi-layer dividend strategy. A piecewise integro-differential equation for the Gerber-Shiu function is derived and solved. Asymptotic formulas of the ruin probability are obtained when the claim size distributions are heavy-tailed.     

On the compound Poisson risk model with dependence based on a generalized Farlie-Gumbel-Morgenstern copula

In this paper we consider an extension to the classical compound Poisson risk model in which we introduce a dependence structure between the claim amounts and the interclaim time. This structure is embedded via a generalized Farlie-Gumbel-Morgenstern copula. In this framework, we derive the Laplace transform of the Gerber-Shiu discounted penalty function. An explicit expression for the Laplace transform of the time of ruin is given for exponential claim sizes.     

On a compound Poisson risk model with delayed claims and random incomes

The main focus of this paper is to analyze the Gerber-Shiu penalty function of a compound Poisson risk model with delayed claims and random incomes. It is assumed that every main claim will produce a by-claim which can be delayed with a certain probability. We derive the integral equation satisfied by the Gerber-Shiu penalty function. Given that the premium size is exponentially distributed, the explicit expression for the Laplace transform of the Gerber-Shiu penalty function is derived. Finally, when the premium sizes have rational Laplace transforms, we also obtain the Laplace transform of the Gerber-Shiu penalty function.     

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.     

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.     

The perturbed compound Poisson risk model with two-sided jumps

In this paper, we consider a perturbed compound Poisson risk model with two-sided jumps. The downward jumps represent the claims following an arbitrary distribution, while the upward jumps are also allowed to represent the random gains. Assuming that the density function of the upward jumps has a rational Laplace transform, the Laplace transforms and defective renewal equations for the discounted penalty functions are derived, and the asymptotic estimate for the probability of ruin is also studied for heavy-tailed downward jumps. Finally, some explicit expressions for the discounted penalty functions, as well as numerical examples, are given.     

Constant dividend barrier in a risk model with interclaim-dependent claim sizes

The risk model with interclaim-dependent claim sizes proposed by Boudreault et al. [Boudreault, M., Cossette, H., Landriault, D., Marceau, E., 2006. On a risk model with dependence between interclaim arrivals and claim sizes. Scand. Actur. J., 265-285] is studied in the presence of a constant dividend barrier. An integro-differential equation for some Gerber-Shiu discounted penalty functions is derived. We show that its solution can be expressed as the solution to the Gerber-Shiu discounted penalty function in the same risk model with the absence of a barrier and a combination of two linearly independent solutions to the associated homogeneous integro-differential equation. Finally, we analyze the expected present value of dividend payments before ruin in the same class of risk models. An homogeneous integro-differential equation is derived and then solved. Its solution can be expressed as a different combination of the two fundamental solutions to the homogeneous integro-differential equation associated to the Gerber-Shiu discounted penalty function.     

On the Markov-dependent risk model with tax

In this paper we consider the Markov-dependent risk model with tax payments in which the claim occurrence, the claim amount as well as the tax rate are controlled by an irreducible discrete-time Markov chainSystems of integro-differential equations satisfied by the expected discounted tax payments and the non-ruin probability in terms of the ruin probabilities under the Markov-dependent risk model without tax are establishedThe analytical solutions of the systems of integro-differential equations are also obtained by the iteration method.     

On a compound Poisson risk model with dependence and in the presence of a constant dividend barrier

In this paper, we consider a classical risk process with dependence and in the presence of a constant dividend barrier. The dependence structure between the claim amounts and the interclaim times is introduced through a Farlie–Gumbel–Morgenstern copula. We analyze the expectation of the discounted penalty function and the expectation of the present value of the distributed dividends. For each function, an integro‐differential equation with boundary conditions is derived, and the solution is provided. Finally, we find an explicit solution for each function when the claim amounts are exponentially distributed. We illustrate the impact of the dependence on these two quantities. Copyright © 2012 John Wiley & Sons, Ltd.     

On a risk model with random incomes and dependence between claim sizes and claim intervals

In this paper, we construct a risk model with a dependence setting where there exists a specific structure among the time between two claim occurrences, premium sizes and claim sizes. Given that the premium size is exponentially distributed, both the Laplace transforms and defective renewal equations for the expected discounted penalty functions are obtained. Exact representations for the solutions of the defective renewal equations are derived through an associated compound geometric distribution. When the claims are subexponentially distributed, the asymptotic formulae for ruin probabilities are obtained. Finally, when the individual premium sizes have rational Laplace transforms, the Laplace transforms for the expected discounted penalty functions are obtained.     

Analysis of the Gerber-Shiu function and dividend barrier problems for a risk process with two classes of claims

In this paper we consider a risk model with two independent classes of insurance risks. We assume that the two independent claim counting processes are, respectively, the Poisson and the generalized Erlang(2) process. We prove that the Gerber-Shiu function satisfies some defective renewal equations. Exact representations for the solutions of these equations are derived through an associated compound geometric distribution and an analytic expression for this quantity is given when the claim severities have rationally distributed Laplace transforms. Further, the same risk model is considered in the presence of a constant dividend barrier. A system of integro-differential equations with certain boundary conditions for the Gerber-Shiu function is derived and solved. Using systems of integro-differential equations for the moment-generating function as well as for the arbitrary moments of the discounted sum of the dividend payments until ruin, a matrix version of the dividends-penalty is derived. An extension to a risk model when the two independent claim counting processes are Poisson and generalized Erlang(ν), respectively, is considered, generalizing the aforementioned results.     

A note on the perturbed compound poisson risk model with a threshold dividend strategy

In this paper, we consider the Perturbed Compound Poisson Risk Model with a threshold dividend strategy （PCT）. Integro-differential equations （IDE） for its Cerber-Shiu functions and dividend payments function are stated. We maily focus on deriving the boundary conditions to solve these equations.     

带注资的二维复合泊松模型的最优分红

研究建立两类理赔关系的二维复合泊松模型的最优分红与注资问题,目标为最大化分红减注资的折现. 该问题由随机控制问题刻画, 通过解相应的哈密尔顿-雅克比-贝尔曼(HJB)方程,得到了最优分红策略,并在指数理赔时明确地解决该问题.