On optimal dividends: From reflection to refraction

The problem goes back to a paper that Bruno de Finetti presented to the International Congress of Actuaries in New York (1957). In a stock company that is involved in risky business, what is the optimal dividend strategy, that is, what is the strategy that maximizes the expectation of the discounted dividends (until possible ruin) to the shareholders? Jeanblanc-Picqué and Shiryaev [Russian Math. Surveys 20 (1995) 257–277] and Asmussen and Taksar [Insurance: Math. Econom. 20 (1997) 1–15] solved the problem by modeling the income process of the company by a Wiener process and imposing the condition of a bounded dividend rate. Here, we present some down-to-earth calculations in this context.