Abstract: The ability of insurance companies to pay dividends is an important indicator that reflects the company’s operating conditions and economic strength. What kind of dividend strategy is adopted affects not only the interests of shareholders, but also the stability of the company’s surplus, the liquidity of assets, the ability to debt, and even the survival of the company. Seeking the optimal dividend strategy is of course one of the important issues that the industry and theorists care about, and it is an important means of corporate risk management. DE FINETTI^[1] proposed a realistic and economically motivated stability criterion: The management of the company should look for maximizing the expectation of the present value of all dividends paid to the shareholders up to ruin time. GERBER^[2] first showed that an optimal dividend strategy has a band structure for the compound Poisson risk model. Since then the optimal dividend problem has been developed rapidly for the compound Poisson risk model. Due to its practical importance, the issue of absolute ruin problem has attracted attention in the actuarial literature. Ruin (negative reserve) does not mean the end of the game but only the necessity of raising additional money and that it will be a good investment to rescue a company when the situation is not too serious. We then assume that when the surplus is negative or the company is on deficit, the company could borrow money at a debit interest force β>0. Meanwhile, the company will repay its debt continuously from its income. Thus, the surplus of the company is driven under the debit interest force β>0, when the surplus is negative. We allow a company to continue its business with debt as long as it is profitable. However, when the surplus of a company is below the level-c/β, we say that absolute ruin occurs at this situation. In the compound Poisson risk model, the surplus cannot return to a positive level once it attains the critical level-c/β. In this case, the value-c/β may be interpreted as the maximum allowable debt for a company. When the surplus is positive, it can be invested to obtain the fixed income. When the transaction costs are taken into account, the optimal dividend strategy becomes more complex. The fixed cost, however small, can have a big effect on the value function. The optimal dividend problem with fixed transaction costs has yielded fruitful results for diffusion processes. However, the works for compound Poisson risk model are relatively rare.
We consider the optimal dividend problem with transaction costs for the compound Poisson model with credit and debit interest and control the times and the amount of dividends to maximize the expected cumulative discounted dividends payment until the time of absolute ruin. Due to the consideration of transaction costs, the problem is formulated as a stochastic impulse control problem. The necessary and sufficient condition for a strategy to be a stationary Markov strategy is presented firstly. The associated measure-valued dynamic programming equation (DPE) is derived by virtue of the theory of measure-valued generators. The verification theorem is proved without additional assumption on the differentiability of the value function. We also show that the optimal dividend strategy constructed in the verification theorem is indeed a stationary Markov strategy. By the Lebesgue decomposition we discuss the relationship between the measure-valued DPE and the QVI satisfied by the value function. We present the existence of the optimal dividend strategies and prove that the optimal strategy is a stationary Markov strategy with a band structure. An algorithm for getting a multi-level lump sum dividend barrier strategy and the corresponding value function is given. The analytical solutions of the value function and the optimal dividend strategy are obtained for the exponential claims.

Key words: optimal dividend problem, transaction costs, measure-valued DPE, Markov strategy

摘要： 本文研究了复合Poisson模型带投资-借贷利率和固定交易费用的最优分红问题。通过控制分红时刻和分红量,最大化直到绝对破产时刻的累积期望折现分红。由于考虑固定交易费用,问题为一个随机脉冲控制问题。首先,本文给出了一个策略是平稳马氏策略的充分必要条件。借助于测度值生成元理论得到测度值动态规划方程 (简称测度值DPE),并且在没有任何附加条件下证明了验证定理。通过Lebesgue分解,本文讨论了测度值DPE和拟变分不等式(简称 QVI)之间的关系,证明了最优分红策略为具有波段结构的平稳马氏策略。最后,本文给出了求解n-波段策略和相应值函数的算法。当索赔额服从指数分布时,得到了值函数的显示解和最优分红策略。

关键词: 最优分红问题, 固定交易费用, 测度值DPE, 马氏策略