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 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.     

常数红利边界下的一类马氏风险模型

本文研究了常数红利边界下一类马氏风险模型的红利派发矩,破产前所有红利的分布等相关问题.利用更新方法,给出了该模型破产前红利折现的期望满足的微分-积分方程,得到破产前所有红利的分布.通过构造特殊的初始条件,得到了相关的方程组解,推广了文献[3]的结果.     

The perturbed Sparre Andersen model with a threshold dividend strategy

In this paper, we consider a Sparre Andersen model perturbed by diffusion with generalized Erlang(n)-distributed inter-claim times and a threshold dividend strategy. Integro-differential equations with certain boundary conditions for the moment-generation function and the mth 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 where the inter-claim times are Erlang(2) distributed and the claim size distribution is exponential is considered in some details.     

一类混合分红策略下的广义Erlang(n)风险模型

本文考虑混合分红策略下索赔来到间隔为广义Erlang(n)分布的更新风险模型,利用指数分布的无记忆性,分别得到破产前期望折现分红函数和折现分红的矩母函数满足的积分-微分方程及其边界条件.最后给出索赔为指数分布及索赔来到间隔为广义Erlang(2)分布的风险模型的期望折现分红函数的精确表达式.     

An elementary approach to discrete models of dividend strategies

The paper studies a discrete counterpart of Gerber et al. (2006). The surplus of an insurance company (before dividends) is modeled as a time-homogeneous Markov chain with possible changes of size +1,0,−1,−2,−3,…. If a barrier strategy is applied for paying dividends, it is shown that the dividends-penalty identity holds. The identity expresses the expected present value of a penalty at ruin in terms of the expected discounted dividends until ruin and the expected present value of the penalty at ruin if no dividends are paid. For the problem of maximizing the difference between the expected discounted dividends until ruin and the expected present value of the penalty at ruin, barrier strategies play a prominent role. In some cases an optimal dividend barrier exists. The paper discusses in detail the special case where the distribution of the change in surplus does not depend on the current surplus (so that in the absence of dividends the surplus process has independent increments). A closed-form result for zero initial surplus is given, and it is shown how the relevant quantities can be calculated recursively. Finally, it is shown how optimal dividend strategies can be determined; typically, they are band strategies.     

常红利边界复合二项模型红利期望现值

在完全离散的复合二项风险模型基础上,考虑常红利边界策略下的红利支付问题．通过两种不同的方法,得到了红利期望现值所满足的两个方程．由这些方程特殊性质,在比较宽松的条件下,通过建立相应的迭代过程,求解出了直到破产发生时红利期望现值的近似值．     

The compound Poisson process perturbed by a diffusion with a threshold dividend strategy

In this paper, we consider the compound Poisson process perturbed by a diffusion in the presence of the so‐called threshold dividend strategy. Within this framework, we prove the twice continuous differentiability of the expected discounted value of all dividends until ruin. We also derive integro‐differential equations for the expected discounted value of all dividends until ruin and obtain explicit expressions for the solution to the equations. Along the same line, we establish explicit expressions for the Laplace transform of the time of ruin and the Laplace transform of the aggregate dividends until ruin. In the case of exponential claims, some examples are provided. Copyright © 2008 John Wiley & Sons, Ltd.     

阈值分红策略下带常利率的对偶风险模型

研究了常利率下基于对偶复合泊松模型带阈值的分红策略,给出了公司在破产时累积红利期望现值函数的两个积分-微分方程,分情况讨论了收益服从指数分布时的显示表达式,以及服从一般分布时的拉普拉斯变换表达式.     

Optimal choice of dividend barriers for a risk process with stochastic return on investments

We consider a risk process with stochastic return on investments and we are interested in expected present value of all dividends paid until ruin occurs when the company uses a simple barrier strategy, i.e. when it pays dividends whenever its surplus reaches a level b. It is shown that given the barrier b, this expected value can be found by solving a boundary value problem for an integro-differential equation. The solution is then found in two special cases; when return on investments is constant and the surplus generating process is compound Poisson with exponentially distributed claims, and also when both return on investments as well as the surplus generating process are Brownian motions with drift. Also in this latter case we are able to find the optimal barrier b^*, i.e. the barrier that gives the highest expected present value of dividends. Parallell with this we treat the problem of finding the Laplace transform of the distribution of the time to ruin when a barrier strategy is employed, noting that the probability of eventual ruin is 1 in this case. The paper ends with a short discussion of the same problems when a time dependent barrier is employed.     

Expected present value of total dividends in a delayed claims risk model under stochastic interest rates

In this paper, a compound binomial risk model with a constant dividend barrier under stochastic interest rates is considered. Two types of individual claims, main claims and by-claims, are defined, where every by-claim is induced by the main claim and may be delayed for one time period with a certain probability. In the evaluation of the expected present value of dividends, the interest rates are assumed to follow a Markov chain with finite state space. A system of difference equations with certain boundary conditions for the expected present value of total dividend payments prior to ruin is derived and solved. Explicit results are obtained when the claim sizes are K_n distributed or the claim size distributions have finite support. Numerical results are also provided to illustrate the impact of the delay of by-claims on the expected present value of dividends.     

具有分红、流动储备金和利率的风险模型的绝对破产

建立了阈值分红策略下具有流动储备金、投资利率和贷款利率的复合泊松风险模型.利用全概率公式和泰勒展式,推导出了该模型的Gerber-Shiu函数和绝对破产时刻的累积分红现值期望满足的积分-微分方程及边界条件,借助Volterra方程,给出了Gerber-Shiu函数的解析表达式.     

Dividends in finite time horizon

In this paper, we consider the classical risk model modified in two different ways by the inclusion of a dividend barrier. For Model I, we present numerical algorithms, which can be used to approximate or bound the expected discounted value of dividends up to a finite time horizon, t, or ruin if this occurs earlier. We extend this by requiring the shareholders to provide the initial capital and to pay the deficit at ruin each time it occurs so that the process then continues after ruin up to time t. For Model I, we assume the full premium income is paid as dividends whenever the surplus exceeds a set level. In our Model II, we assume dividends are paid at a rate less than the rate of premium income. Copyright © 2012 John Wiley & Sons, Ltd.     

Optimal dividend and dynamic reinsurance strategies with capital injections and proportional costs

We consider an optimization problem of an insurance company in the diffusion setting, which controls the dividends payout as well as the capital injections. To maximize the cumulative expected discounted dividends minus the penalized discounted capital injections until the ruin time, there is a possibility of (cheap or non-cheap) proportional reinsurance. We solve the control problems by constructing two categories of suboptimal models, one without capital injections and one with no bankruptcy by capital injection. Then we derive the explicit solutions for the value function and totally characterize the optimal strategies. Particularly, for cheap reinsurance, they are the same as those in the model of no bankruptcy.     

EXPECTED PRESENT VALUE OF TOTAL DIVIDENDS IN THE COMPOUND BINOMIAL MODEL WITH DELAYED CLAIMS AND RANDOM INCOME

In this paper, a compound binomial model with a constant dividend barrier and random income is considered. Two types of individual claims, main claims and by-claims, are defined, where every by-claim is induced by the main claim and may be delayed for one time period with a certain probability. The premium income is assumed to another binomial process to capture the uncertainty of the customer＇s arrivals and payments. A system of difference equations with certain boundary conditions for the expected present value of total dividend payments prior to ruin is derived and solved. Explicit results are obtained when the claim sizes are Kn distributed or the claim size distributions have finite support. Numerical results are also provided to illustrate the impact of the delay of by-claims on the expected present value of dividends.     

Dividend Maximization when Cash Reserves Follow a Jump-diffusion Process

This paper deals with the dividend optimization problem for an insurance company, whose surplus follows a jump-diffusion process. The objective of the company is to maximize the expected total discounted dividends paid out until the time of ruin. Under concavity assumption on the optimal value function, the paper states some general properties and, in particular, smoothness results on the optimal value function, whose analysis mainly relies on viscosity solutions of the associated Hamilton-Jacobi-Bellman （HJB） equations. Based on these properties, the explicit expression of the optimal value function is obtained. And some numerical calculations are presented as the application of the results.     

Optimal dividend strategy in compound binomial model with bounded dividend rates

We consider the compound binomial model, and assume that dividends are paid to the shareholders according to an admissible strategy with dividend rates bounded by a constant.The company controls the amount of dividends in order to maximize the cumulative expected discounted dividends prior to ruin. We show that the optimal value function is the unique solution of a discrete HJB equation. Moreover, we obtain some properties of the optimal payment strategy, and offer a simple algorithm for obtaining the optimal strategy. The key of our method is to transform the value function. Numerical examples are presented to illustrate the transformation method.     

Parisian ruin with a threshold dividend strategy under the dual Lévy risk model

We consider the threshold dividend strategy where a company’s surplus process is described by the dual Lévy risk model. Namely, the company chooses to pay dividends at a constant rate only when the surplus is above some nonnegative threshold. Classically, such a company is referred to be ruined immediately when the surplus level becomes negative. Recently, researchers investigate the Parisian ruin problem where the company is allowed to operate under negative surplus for a predetermined period known as the Parisian delay. With the help of the fluctuation identities of spectrally negative Lévy processes, we obtain an explicit expression of the expected discounted dividends until Parisian ruin in terms of the relevant scale functions and certain probabilities that need to be evaluated for each specific Lévy process. The optimal threshold level under such a threshold dividend strategy is deduced. Applications and numerical examples are given to illustrate the theoretical results and examine how the expected discounted aggregate dividends and the optimal threshold level change in response to different Parisian delays.     

Asymptotically optimal dividend policy for regime-switching compound Poisson models

This work develops asymptotically optimal dividend policies to maximize the expected present value of dividends until ruin.Compound Poisson processes with regime switching are used to model the surplus and the switching（a continuous-time controlled Markov chain） represents random environment and other economic conditions.Assuming the switching to be fast varying together with suitable conditions,it is shown that the system has a limit that is an average with respect to the invariant measure of a related Markov chain.Under simple conditions,the optimal policy of the limit dividend strategy is a threshold policy.Using the optimal policy of the limit system as a guide,feedback control for the original surplus is then developed.It is demonstrated that the constructed dividend policy is asymptotically optimal.     

方差保费准则下的最优脉冲控制

本文研究保险公司在有再保险控制下的最优脉冲分红问题. 对保险公司的理赔损失, 假定有两家再保险公司参与分保, 且保险公司与两家再保险公司采取不同参数下的方差保费准则. 进一步, 假定保险公司有股东红利分配, 且每次分红有固定交易费和比例税收, 即脉冲分红. 在扩散逼近模型下, 本文应用随机动态规划方法研究破产前的最大期望折现分红, 给出值函数的解析表达式, 进而获得最优再保险策略和分红策略的具体形式.