带常利率的Erlang(2)风险模型的罚金折现期望函数

The purpose of this paper is to consider the expected value of a discounted penalty due at ruin in the Erlang（2） risk process under constant interest force. An integro-differential equation satisfied by the expected value and a second-order differential equation for the Laplace transform of the expected value are derived. In addition, the paper will present the recursive algorithm for the joint distribution of the surplus immediately before ruin and the deficit at ruin. Finally, by the differential equation, the defective renewal equation and the explicit expression for the expected value are given in the interest-free case.     

THE RISK MODEL OF THE EXPECTED DISCOUNTED PENALTY FUNCTION WITH CONSTANT INTEREST FORCE

In this article, the expected discounted penalty functionΦδ,α(u) with constant interest 5 and "discounted factor" exp(-αTδ) is considered. As a result, the integral equation ofΦδ,α(u) is derived and an exact solution forΦδ,α(0) is found. The relation between the joint density of the surplus immediately prior to ruin, and the deficit at ruin and the density of the surplus immediately prior to ruin is then obtained based on analytical methods.     

The gerber-shiu expected discounted penalty function for Lévy insurance risk processes

In this paper,we study a general Lévy risk process with positive and negative jumps.A renewal equation and an infinite series expression are obtained for the expected discounted penalty function of this risk model.We also examine some asymptotic behaviors for the ruin probability as the initial capital tends to infinity.     

A class of delayed renewal risk processes with a threshold dividend strategy

This paper considers a class of delayed renewal risk processes with a threshold dividend strategy. The main result is an expression of the Gerber-Shiu expected discounted penalty function in the delayed renewal risk model in terms of the corresponding Cerber-Shiu function in the ordinary renewal model. Subsequently, this relationship is considered in more detail in both the stationary renewal risk model and the ruin probability.     

A ruin model with random income and dependence between claim sizes and claim intervals

In this paper,we consider a generalization of the classical ruin model,where the income is random and the distribution of the time between two claim occurrences depends on the previous claim size.This model is more appropriate than the classical ruin model.Explicit expression for the generating function of the Gerber-Shiu expected discounted penalty function are derived.A similar model is discussed.Finally,the result are showed by two examples.     

The Gerber-Shiu discounted penalty function for a compound binomial risk model with by-claims

A recursive formula of the Gerber-Shiu discounted penalty function for a compound binomial risk model with by-claims is obtained. In the discount-free case, an explicit formula is given. Utilizing such an explicit expression, we derive some useful insurance quantities, including the ruin probability, the density of the deficit at ruin, the joint density of the surplus immediately before ruin and the deficit at ruin, and the density of the claim causing ruin.     

The optimal policy for insurance company under consideration of internal competition and the time value of ruin

This paper considers the dividend optimization problem for an insurance company under the consideration of internal competition between different units inside the company. The objective is to find a reinsurance policy and a dividend payment scheme so as to maximize the expected discounted value of the dividend payment, and the expected present value of an amount which the insurer earns until the time of ruin. By solving the corresponding constrained Hamilton-Jacobi-Bellman （HJB） equation, we obtain the value function and the optimal reinsurance policy and dividend payment.     

ON THE EXPECTED DISCOUNTED PENALTY FUNCTION IN A MARKOV-DEPENDENT RISK MODEL WITH CONSTANT DIVIDEND BARRIER

This article considers a Markov-dependent risk model with a constant dividend barrier. A system of integro-differential equations with boundary conditions satisfied by the expected discounted penalty function, with given initial environment state, is derived and solved. Explicit formulas for the discounted penalty function are obtained when the initial surplus is zero or when all the claim amount distributions are from rational family. In two state model, numerical illustrations with exponential claim amounts are given.     

First passage models for denumerable semi-Markov decision processes with nonnegative discounted costs

This paper considers a first passage model for discounted semi-Markov decision processes with denumerable states and nonnegative costs.The criterion to be optimized is the expected discounted cost incurred during a first passage time to a given target set.We first construct a semi-Markov decision process under a given semi-Markov decision kernel and a policy.Then,we prove that the value function satisfies the optimality equation and there exists an optimal(or e-optimal) stationary policy under suitable conditions by using a minimum nonnegative solution approach.Further we give some properties of optimal policies.In addition,a value iteration algorithm for computing the value function and optimal policies is developed and an example is given.Finally,it is showed that our model is an extension of the first passage models for both discrete-time and continuous-time Markov decision processes.     

On the Gerber-Shiu discounted penalty function for a surplus process described by PDMPs

In this paper, we investigate the Gerber-Shiu discounted penalty function for the surplus process described by a piecewise deterministic Markov process （PDMP）. We derive an integral equation for the Gerber-Shiu discounted penalty function, and obtain the exact solution when the initial surplus is zero. Dickson formulae are also generalized to the present surplus process.     

稀疏风险模型的期望折扣罚金函数

本文考虑了一类风险模型,其中保费到达过程是一个参数为$\lambda>0$的Poisson过程,而理赔过程是保费到达过程的稀疏过程. 在该模型下,我们得到了期望折扣罚金函数所满足的积分方程,积分--微分方程以及递推公式, 并且当保费和理赔额均为指数分布时,我们使用积分--微分方程获得了破产时刻的Laplace变换和在破产时刻的赤字的闭式表达式.     

随机保费风险模型下的平均折现罚金函数

本文研究随机保费风险模型下与破产时刻相关的平均折现罚金函数. 与经典的Cram\'{e}r-Lundberg模型相比这里的保费过程不再是时间的线性函数, 而是一个与理赔独立的复合Possion过程. 我们得到了罚金函数所满足的积分方程, 它提供了一种研究破产量的统一方法. 利用该积分方程我们得到了破产时刻, 破产时赤字, 破产前瞬时盈余的Laplace变换; 并在指数分布的特殊情况下求出了他们的显著表达式, 推广了Boikov (2003)的结论.     

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.     

带常数边界的平衡更新风险模型的破产问题

本文讨论带常数边界的平衡更新风险模型的破产问题.利用Markov性质,给出惩罚函数满足的积分-微分方程,证明其惩罚函数可由更新风险模型的惩罚函数表示,并且给出一个具体的例子.     

保费随机的离散风险模型的罚金期望函数

本文把经典的复合二项风险模型进行推广,其中保费收取方式不再是时间的线性函数而是一个二项过程.我们把它的罚金期望看成初始资本的函数,得到了罚金期望函数的递推公式和渐近估计,最后利用罚金期望函数的递推公式和渐近估计给出了几个重要的破产量的递推公式及其渐近估计.     

The expected discounted penalty function under a risk model with stochastic income

Quantities of interest in ruin theory are investigated under the general framework of the expected discounted penalty function, assuming a risk model where both premiums and claims follow compound Poisson processes. Both a defective renewal equation and an integral equation satisfied by the expected discounted penalty function are established. Some implications that these equations have on particular quantities such as the discounted deficit and the probability of ultimate ruin are illustrated. Finally, the case when premiums have Erlang(n,β) distribution and the distribution of the claims is arbitrary is investigated in more depth. Throughout the paper specific examples where claims and premiums have particular distributions are provided.     

On the expected discounted penalty function for risk process with tax

In this paper we consider the generalized Cramér-Lundberg risk model including tax payments. We investigate how tax payments affect the behavior of a Cramér-Lundberg surplus process by defining an expected discounted penalty function at ruin. We derive an explicit expression for this function by solving a differential equation. Consequently, the explicit formulas for the discounted probability density function of the surplus immediately before ruin and the discounted joint probability density function of the surplus immediately before ruin and the deficit at ruin are obtained. We also give explicit expressions for the function for exponential claims.     

On the expected discounted penalty function at ruin of a surplus process with interest

In this paper, we study the expected value of a discounted penalty function at ruin of the classical surplus process modified by the inclusion of interest on the surplus. The ‘penalty’ is simply a function of the surplus immediately prior to ruin and the deficit at ruin. An integral equation for the expected value is derived, while the exact solution is given when the initial surplus is zero. Dickson’s [Insurance: Mathematics and Economics 11 (1992) 191] formulae for the distribution of the surplus immediately prior to ruin in the classical surplus process are generalised to our modified surplus process.     

The Sparre Andersen Risk Process with Investment and Debit Interest

In this paper, we study absolute ruin problems for the Sparre Andersen risk process with generalized Erlang()-distributed inter-claim times, investment and debit interest. We first give a system of integro-differential equations with certain boundary conditions satisfied by the expected discounted penalty function at absolute ruin. Second, we obtain a defective renewal equation under some special cases, then based on the defective renewal equation we derive 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 for generalized Erlang(2) inter-claim times and exponential claims.