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.     

On the Expected Present Value of Total Dividends in a Risk Model with Potentially Delayed Claims

In this paper, we consider a risk model in which two types of individual claims, main claims and by-claims, are defined. Every by-claim is induced by the main claim randomly and may be delayed for one time period with a certain probability. The dividend policy that certain amount of dividends will be paid as long as the surplus is greater than a constant dividend barrier is also introduced into this delayed claims risk model. By means of the probability generating functions, formulae for the expected present value of total dividend payments prior to ruin are obtained for discrete-type individual claims. Explicit expressions for the corresponding results are derived for K n claim amount distributions. Numerical illustrations are also given.     

具有常红利边界和延迟索赔的一类离散更新风险模型

考虑了具有常红利边界和延迟索赔的一类离散更新风险模型,其中间隔索赔到达时间从离散phase-type分布.定义了两种类型的索赔:主索赔和副索赔,主索赔以一定的概率引起副索赔且副索赔会以一定的概率被延迟到下一时段.通过引入辅助风险模型,推导了破产前红利折现期望满足的差分方程及其解.最后给出了当索赔额服从几何分布时的有关数值例子.     

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.     

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.     

Optimal dividend payout for classical risk model with risk constraint

In this paper we consider the problem of maximizing the total discounted utility of dividend payments for a Cramér-Lundberg risk model subject to both proportional and fixed transaction costs.We assume that dividend payments are prohibited unless the surplus of insurance company has reached a level b.Given fixed level b,we derive a integro-differential equation satisfied by the value function.By solving this equation we obtain the analytical solutions of the value function and the optimal dividend strategy when claims are exponentially distributed.Finally we show how the threshold b can be determined so that the expected ruin time is not less than some T.Also,numerical examples are presented to illustrate our results.     

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 classical risk model with credit and debit interests under absolute ruin

In this paper, we consider the dividend payments in a compound Poisson risk model with credit and debit interests under absolute ruin. We first obtain the integro-differential equations satisfied by the moment generating function and moments of the discounted aggregate dividend payments. Secondly, applying these results, we get the explicit expressions of them for exponential claims. Then, we give the numerical analysis of the optimal dividend barrier and the expected discounted aggregate dividend payments which are influenced by the debit and credit interests. Finally, we find the integro-differential equations satisfied by the Laplace transform of absolute ruin time and give its explicit expressions when the claim sizes are exponentially distributed.     

Statistical estimation for some dividend problems under the compound Poisson risk model

In this paper, we consider some dividend problems in the classical compound Poisson risk model under a constant barrier dividend strategy. Suppose that the Poisson intensity for the claim number process and the distribution for the individual claim sizes are both unknown. We use the COS method to study the statistical estimation for the expected present value of dividend payments before ruin and the expected discounted penalty function. The convergence rates under large sample setting are derived. Some simulation results are also given to show effectiveness of the estimators under finite sample setting.     

Dividend payments in the classical risk model under absolute ruin with debit interest

This paper attempts to study the dividend payments in a compound Poisson surplus process with debit interest. Dividends are paid to the shareholders according to a barrier strategy. An alternative assumption is that business can go on after ruin, as long as it is profitable. When the surplus is negative, a debit interest is applied. At first, we obtain the integro‐differential equations satisfied by the moment‐generating function and moments of the discounted dividend payments and we also prove the continuous property of them at zero. Then, applying these results, we get the explicit expressions of the moment‐generating function and moments of the discounted dividend payments for exponential claims. Furthermore, we discuss the optimal dividend barrier when the claim sizes have a common exponential distribution. Finally, we give the numerical examples for exponential claims and Erlang (2) claims. Copyright © 2008 John Wiley & Sons, Ltd.     

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．     

Dividend maximization under consideration of the time value of ruin

In the Cramér-Lundberg model and its diffusion approximation, it is a classical problem to find the optimal dividend payment strategy that maximizes the expected value of the discounted dividend payments until ruin. One often raised disadvantage of this approach is the fact that such a strategy does not take the lifetime of the controlled process into account. In this paper we introduce a value function which considers both expected dividends and the time value of ruin. For both the diffusion model and the Cramér-Lundberg model with exponential claim sizes, the problem is solved and in either case the optimal strategy is identified, which for unbounded dividend intensity is a barrier strategy and for bounded dividend intensity is of threshold type.     

The Optimal Dividend and Capital Injection Strategies in the Classical Risk Model with Randomized Observation Periods

This paper considers the optimal dividend and capital injection strategies in the classical risk model with randomized observation periods. Assume that ruin is prohibited. We aim to maximise the expected discounted dividend payments minus the expected penalised discounted capital injections. We derive the associated Hamilton-Jacobi-Bellman (HJB) equation and prove the verification theorem. The optimal control strategy and the optimal value function are obtained under the assumption that the claim sizes are exponentially distributed.     

Optimal dividends in the Brownian motion risk model with interest

In this paper, we consider a Brownian motion risk model, and in addition, the surplus earns investment income at a constant force of interest. The objective is to find a dividend policy so as to maximize the expected discounted value of dividend payments. It is well known that optimality is achieved by using a barrier strategy for unrestricted dividend rate. However, ultimate ruin of the company is certain if a barrier strategy is applied. In many circumstances this is not desirable. This consideration leads us to impose a restriction on the dividend stream. We assume that dividends are paid to the shareholders according to admissible strategies whose dividend rate is bounded by a constant. Under this additional constraint, we show that the optimal dividend strategy is formed by a threshold strategy.     

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.     

On the renewal risk model under a threshold strategy

In this paper, we consider the renewal risk process under a threshold dividend payment strategy. For this model, the expected discounted dividend payments and the Gerber–Shiu expected discounted penalty function are investigated. Integral equations, integro-differential equations and some closed form expressions for them are derived. When the claims are exponentially distributed, it is verified that the expected penalty of the deficit at ruin is proportional to the ruin probability.     

The maximum surplus before ruin and related problems in a jump-diffusion renewal risk process

In this paper, we investigate a Sparre Andersen risk model perturbed by diffusion with phase-type inter-claim times. We mainly study the distribution of maximum surplus prior to ruin. A matrix form of integro-differential equation for this quantity is derived, and its solution can be expressed as a linear combination of particular solutions of the corresponding homogeneous integro-differential equations. By using the divided differences technique and nonnegative real part roots of Lundberg’s equation, the explicit Laplace transforms of particular solutions are obtained. Specially, we can deduce closed-form results as long as the individual claim size is rationally distributed. We also give a concise matrix expression for the expected discounted dividend payments under a barrier dividend strategy. Finally, we give some examples to present our main results.     

The phase-type risk model perturbed by diffusion under a threshold dividend strategy

This paper considers a perturbed renewal risk process in which the inter-claim times have a phase-type distribution under a threshold dividend strategy. Integro-differential equations with certain boundary conditions for the moment-generating function and the mth moment of the present value of all dividends until ruin are derived. Explicit expressions for the expectation of the present value of all dividends until ruin are obtained when the claim amount distribution is from the rational family. Finally, we present an example.     

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In the classical Cram\'{e}r-Lundberg model in risk theory the problem of finding the optimal dividend strategy and optimal dividend return function is a widely discussed topic. In the present paper, we discuss the problem of maximizing the expected discounted net dividend payments minus the expected discounted costs of injecting new capital, in the Cram\'{e}r-Lundberg model with proportional taxes and fixed transaction costs imposed each time the dividend is paid out and with both fixed and proportional transaction costs incurred each time the capital injection is made. Negative surplus or ruin is not allowed. By solving the corresponding quasi-variational inequality, we obtain the analytical solution of the optimal return function and the optimal joint dividend and capital injection strategy when claims are exponentially distributed.     

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.