Optimal dividend strategy for the dual model with surplus-dependent expense

In this paper, we consider the optimal dividend problem for the dual model with surplus-dependent expense. The objective is to find the optimal dividend-payment strategy that maximizes the expected discounted value of dividends until the time of ruin. We give the analytic characterizations of admissible strategy and Markov strategy and show that a dividend strategy is a Markov strategy if and only if it is an additive functional of the controlled surplus process. We use the theory of measure-valued generators to derive a measure-valued dynamic programming equation (DPE). And the verification theorem is proved without additional assumptions on the regularity of the value function. Under the assumption that the value function is locally Lipschitz continuous, the optimal strategy is presented as a Markov strategy and hence a band strategy. Numerical examples are also given.