Abstract：In this paper, we extend the optimal dividend and capital injection problem with affine penalty at ruin in (Xu, R. and Woo, J.K. Insurance: Mathematics and Economics 92:1–16 (2020)) 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. 

报告人简介：徐冉，西交利物浦大学金融与精算数学系助理教授，2018 年获得香港大学统计精算系博士学位，2018-2019 年在加拿大 Concordia 大学从事博士后研究工作，2019 年 9 月加入西交利物浦大学，主要研究方向为保险精算、风险理论及金融保险中的随机最优化问题。