Stochastic Control of Dividends with a Drawdown Penalty

We consider a diffusion risk model where dividends are paid at rate U(t) ∈ [0, u₀]. We are interested in maximising the dividend payments under a drawdown constraint, that is, we penalise a drawdown size larger than a level $d > 0$. We show that the optimal dividend rate $U(t)$ is either zero or the maximal rate u₀ and determine the optimal strategy. Moreover, we derive an explicit expression for the value function by solving a system of differential equations.