Optimal periodic dividend and capital injection problem for spectrally positive Lévy processes

In this paper, we investigate an optimal periodic dividend and capital injection problem for spectrally positive Lévy processes. We assume that the periodic dividend strategy has exponential inter-dividend-decision times and continuous monitoring of solvency. Both proportional and fixed transaction costs from capital injection are considered. The objective is to maximize the total value of the expected discounted dividends and the penalized discounted capital injections until the time of ruin. By the fluctuation theory of Lévy processes in Albrecher et al. (2016), the optimal periodic dividend and capital injection strategies are derived. We also find that the optimal return function can be expressed in terms of the scale functions of Lévy processes. Finally, numerical examples are studied to illustrate our results.     

复合Poisson 模型带比例与固定交易费用的最优分红与注资

研究了复合Poisson 模型带比例与固定费用的最优分红与注资问题. 每次分红与注资时, 存在比例及固定的交易费用. 通过控制分红与注资的时刻以及分红及注资量,实现破产前分红减注资的折现期望的最大化. 由于存在固定交易费用, 问题为一个脉冲控制问题. 根据问题的参数不同, 问题的解可分为两大类. 一类解为只进行最优分红不需要注资, 而另一类情况需要注资. 需要注资时, 最优注资策略由最优注资上界以及最优注资下界描述. 当赤字小于最优注资下界的绝对值时, 进行注资. 最后, 在理赔为指数分布时明确地给出了两类共七种最优策略以及值函数的形式. 从而彻底地解决了该问题.     

Optimal dividend and capital injection problem with a random time horizon and a ruin penalty in the dual model

In the dual risk model, we consider the optimal dividend and capital injection problem, which involves a random time horizon and a ruin penalty. Both fixed and proportional costs from the transactions of capital injection are considered. The objective is to maximize the total value of the expected discounted dividends, and the penalized discounted both capital injections and ruin penalty during the horizon, which is described by the minimum of the time of ruin and an exponential random variable. The explicit solutions for optimal strategy and value function are obtained, when the income jumps follow a hyper-exponential distribution.Besides, some numerical examples are presented to illustrate our results.     

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??In this paper, we consider the optimal dividend problem in the spectrally positive L\'{e}vy model with regime switching. By an auxiliary optimal problem, the principle of dynamic programming and the fluctuation theory of L\'{e}vy processes, we show that optimal strategy is a modulated barrier strategy. The value function and the optimal dividend barrier are obtained by iteration.     

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.     

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 .     

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.     

Optimal control problem for an insurance surplus model with debt liability

For an insurance company with a debt liability, they could make some management actions, such as reinsurance, paying dividends, and capital injection, to balance the profitability and financial bankruptcy. Our objective is to determine risk retention rate, dividend, and capital injection strategy so as to maximize the expected discounted dividends minus the discounted cost of capital injection until the time of ruin. We assume that the dividend payments and capital injection should occur with both fixed and proportional costs. We obtain explicit expressions of the optimal value functions as well as the corresponding optimal joint strategies by routine procedures in a comprehensive basic model using a new technique to solve the related equations. Our results show that whether recapitalizing is profitable or not depends on the costs of capital raising and that the firm injects capital only when the reserves are zero and recapitalizes to the optimal reserves level if the cost of external capital is low. Copyright © 2013 John Wiley & Sons, Ltd.     

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.     

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.