In this paper, we extend the optimal dividend and capital injection problem with affine penalty at ruin in (Xu, R. & Woo, J.K. (2020). Insurance: Mathematics and Economics 92: 1–16) to the case with singular dividend payments. The asymptotic relationships between our value function to the one with bounded dividend density are studied, which also help to verify that our value function is a viscosity solution to the associated Hamilton–Jacob–Bellman Quasi-Variational Inequality (HJBQVI). We also show that the value function is the smallest viscosity supersolution within certain functional class. A modified comparison principle is proved to guarantee the uniqueness of the value function as the viscosity solution within the same functional class. Finally, a band-type dividend and capital injection strategy is constructed based on four crucial sets; and the optimality of such band-type strategy is proved by using fixed point argument. Numerical examples of the optimal band-type strategies are provided at the end when the claim size follows exponential and gamma distribution, respectively.

在本文中，我们扩展了破产时仿射惩罚的最优股息和注资问题（Xu, R. & Woo, JK (2020)。保险：数学与经济学92: 1-16) 到单一股息支付的情况。研究了我们的价值函数与有界红利密度函数之间的渐近关系，这也有助于验证我们的价值函数是相关 Hamilton-Jacob-Bellman 拟变分不等式 (HJBQVI) 的粘性解。我们还表明，价值函数是特定功能类别中最小的粘度超解。证明了改进的比较原理可以保证值函数作为同一函数类中粘度解的唯一性。最后，基于四个关键集构建了波段式分红注资策略；并通过不动点论证证明了这种带型策略的最优性。