Upper and lower bounds for dividends in the discrete model

A discrete model of the capital of an insurance company applying a barrier strategy is considered. Properties of the expected discounted dividends until ruin are studied. Assuming that the company can function after ruin and its shareholders cover the deficit and raise the capital up to some positive level, the total expected discounted profit is studied. Upper and lower bounds for dividends and profit are obtained.     

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.     

Obtaining the dividends-penalty identities by interpretation

The dividends-penalty identity is a relation between three functions: the discounted penalty function without dividends, the discounted penalty function if a barrier dividend strategy is applied, and the expected discounted dividends until ruin. The classical model of risk theory is modified in that the deterministic premiums are replaced by a compound Poisson process with exponential jumps. In this model, the dividends-penalty identity is new and can be derived by interpretation. Then the dividends-penalty identity in the classical model is obtained as a limit.     

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.     

On the Gerber-Shiu Function and Optimal Dividend Strategy for a Thinning Risk Model

In this paper, the risk model under constant dividend barrier strategy is studied, in which the premium income follows a compound Poisson process and the arrival of the claims is a p-thinning process of the premium arrival process. The integral equations with boundary conditions for the expected discounted aggregate dividend payments and the expected discounted penalty function until ruin are derived. In addition, the explicit expressions for the Laplace transform of the ruin time and the expected aggregate discounted dividend payments until ruin are given when the individual stochastic premium amount and claim amount are exponentially distributed. Finally, the optimal barrier is presented under the condition of maximizing the expectation of the difference between discounted aggregate dividends until ruin and the deficit at ruin.     

Optimality of the threshold dividend strategy for the compound Poisson model

In this paper, we consider the optimal dividend problem for the compound Poisson risk model. We assume that dividends are paid to the shareholders according to an admissible strategy with dividend rate bounded by a constant. Our objective is to find a dividend policy so as to maximize the expected discounted value of dividends until ruin. We give sufficient conditions under which the optimal strategy is of threshold type.     

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.     

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.     

On a dual model with a dividend threshold

In insurance mathematics, a compound Poisson model is often used to describe the aggregate claims of the surplus process. In this paper, we consider the dual of the compound Poisson model under a threshold dividend strategy. We derive a set of two integro-differential equations satisfied by the expected total discounted dividends until ruin and show how the equations can be solved by using only one of the two integro-differential equations. The cases where profits follow an exponential or a mixture of exponential distributions are then solved and the discussion for the case of a general profit distribution follows by the use of Laplace transforms. We illustrate how the optimal threshold level that maximizes the expected total discounted dividends until ruin can be obtained, and finally we generalize the results to the case where the surplus process is a more general skip-free downwards Lévy process.     

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.     

Optimal Dividend and Capital Injection Strategies with Transaction Costs and Exponentially Distributed Observation Time

In this paper, we consider the optimal joint dividend and capital injection strategy with proportional and fixed costs. It supposes that capitals can be injected whenever they are profitable, but dividends can only be paid at the arrival times of a Poisson process with intensity . Our objective is to determine an optimal strategy of maximizing the expected cumulative discounted dividends minus the expected discounted costs of capital injections before bankruptcy. By solving some impulse problems, we get the closed-form solutions depending on the parameters of model. Some known results in Lokka and Zervos (2008) can be viewed as limiting cases when .     

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.     

The expectation of aggregate discounted dividends for a Sparre Anderson risk process perturbed by diffusion

In this paper, we study the expectation of aggregate dividends until ruin for a Sparre Andersen risk process perturbed by diffusion under a threshold strategy, in which claim waiting times have a common generalized Erlang(n) distribution. For this strategy, we assume that if the surplus is above certain threshold level before ruin, dividends are continuously paid at a constant rate that does not exceed the premium rate, and if not, no dividends are paid. We obtain some integro-differential equations satisfied by the expected discounted dividends, and further its renewal equations. Finally, applying these results to the Erlang(2) risk model perturbed by diffusion, where claims have a common exponential distributions, we give some explicit expressions and numerical analysis. Copyright © 2007 John Wiley & Sons, Ltd.     

Dividend optimization under the gamma-distribution of claims

The classic insurance company work model with gamma-distribution of claim amount is considered. It is supposed that the company applies a dividend barrier strategy. The form of the expected discounted dividends accumulated until the ruin and the expected discounted deficit at the ruin are found. We deal with the optimal barriers which maximize either the dividends amount or shareholders profit. The barrier optimization is illustrated by some examples.     

Dividend problems in the dual model with diffusion and exponentially distributed observation time

Consider dividend problems in the dual model with diffusion and exponentially distributed observation time where dividends are paid according to a barrier strategy. Assume that dividends can only be paid with a certain probability at each point of time, that is, on each observation, if the surplus exceeds the barrier, the excess is paid as dividend. In this paper, integro-differential equations for the expected discounted sum of dividends paid until ruin and the Laplace transform of ruin time are derived. When the gains are exponentially distributed, explicit expressions for the ruin probability, the expected discounted sum of dividends paid until ruin, the Laplace transform of ruin time and the expectation of ruin time are also obtained.     

经典风险模型最优分红值的估计方法

本文考虑经典风险模型在障碍分红策略下的最优分红值的估计问题.当个体索赔额是混合指数分布时,给出最优分红值的解析表达式.但当个体索赔额是一般分布时,最优分红值的解析表达式往往不能得到,这时我们提供了两种估计方法,一是Lundberg渐近估计法,二是离散化模型估计法.最后给出几个数值例子,对不同计算方法下的估计值作出比较.     

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.     

Alternative approach to the optimality of the threshold strategy for spectrally negative Lévy processes

Consider the optimal dividend problem for an insurance company whose uncontrolled surplus precess evolves as a spectrally negative Levy process. We assume that dividends are paid to the shareholders according to admissible strategies whose dividend rate is bounded by a constant. The objective is to find a dividend policy so as to maximize the expected discounted value of dividends which are paid to the shareholders until the company is ruined. In this paper, we show that a threshold strategy （also called refraction strategy） forms an optimal strategy under the condition that the Levy measure has a completely monotone density.     

Constant barrier strategies in a two-state Markov-modulated dual risk model

In this paper,we consider the dividend problem in a two-state Markov-modulated dual risk model,in which the gain arrivals,gain sizes and expenses are influenced by a Markov process.A system of integrodifferential equations for the expected value of the discounted dividends until ruin is derived.In the case of exponential gain sizes,the equations are solved and the best barrier is obtained via numerical example.Finally,using numerical example,we compare the best barrier and the expected discounted dividends in the two-state Markov-modulated dual risk model with those in an associated averaged compound Poisson risk model.Numerical results suggest that one could use the results of the associated averaged compound Poisson risk model to approximate those for the two-state Markov-modulated dual risk model.     

A Markov decision problem in a risk model with interest rate and Markovian environment

We consider the compound binomial model in a Markovian environment presented by Cossette et al.(2004). We modify the model via assuming that the company receives interest on the surplus and a positive real-valued premium per unit time, and introducing a control strategy of periodic dividend payments. A Markov decision problem arises and the control objective is to maximize the cumulative expected discounted dividends paid to the shareholders until ruin minus a discounted penalty for ruin. We show that under the absence of a ceiling of dividend rates the optimal strategy is a conditional band strategy given the current state of the environment process. Under the presence of a ceiling for dividend rates, the character of the optimal control strategy is given. In addition, we offer an algorithm for the optimal strategy and the optimal value function.Numerical results are provided to illustrate the algorithm and the impact of the penalty.