**1. Introduction**

Risk theory initially revolved around minimizing the probability of ruin. However, shareholders are more interested in maximizing the value of the company than minimizing risks. Therefore, (de Finetti 1957) suggested finding the optimal dividend policies which maximize the expected value of the sum of discounted future dividend payments up to the time of ruin; see also (Miller and Modigliani 1961). Another interesting objective, as suggested by (Shreve et al. 1984), is to maximize the expected discounted cumulative dividends while redressing the reserves by injecting capital each time it becomes necessary.

This note is motivated by subsequent results obtained by (Løkka and Zervos 2008; Lindensjö and Lindskog 2019) for a Brownian motion with drift, and by (Zhu and Yang 2016) for diffusions. Their results state that, depending on the size of transaction costs, one of the following strategies is optimal:

