In this paper, we investigate the optimal dividend problem under Parisian ruin with affine penalty payments at Parisian ruin time. The underlying risk process is assumed to be a spectrally negative Lévy risk process. With the help of the dynamic programming principle, we prove that the value function associated to our optimal control problem is the smallest solution with certain characteristics to the corresponding Hamilton–Jacobi–Bellman (HJB) equation. In addition, the form of the performance function under barrier dividend strategy is expressed in terms of various extended scale functions. Then we identify a condition under which the performance function under certain barrier strategy is also a solution to the HJB equation, which in turn illustrates the optimalilty of such barrier dividend strategy among all admissible strategies. Various numerical examples are also given when the underlying risk process is compound Poisson process, Brownian motion with drift and jump-diffusion process.

在本文中，我们研究了在巴黎废墟时期具有仿射罚金支付的最优股息问题。假定潜在的风险过程是频谱负面的Lévy风险过程。借助动态规划原理，我们证明了与最优控制问题相关的价值函数是对相应的汉密尔顿-雅各比-贝尔曼（HJB）方程具有某些特征的最小解。此外，障碍分红策略下的绩效函数形式以各种扩展规模函数表示。然后，我们确定了一个条件，在该条件下，特定障碍策略下的绩效函数也是HJB方程的解，这又说明了所有可接受策略中此类障碍红利策略的最优性。