Optimal dividends in the dual model under transaction costs

We analyze the optimal dividend payment problem in the dual model under constant transaction costs. We show, for a general spectrally positive L{e}vy process, an optimal strategy is given by a $(c_1,c_2)$-policy that brings the surplus process down to $c_1$ whenever it reaches or exceeds $c_2$ for some $0 leq c_1 < c_2$. The value function is succinctly expressed in terms of the scale function. A series of numerical examples are provided to confirm the analytical results and to demonstrate the convergence to the no-transaction cost case, which was recently solved by Bayraktar et al. (2013).

On optimal dividends in the dual model

We revisit the dividend payment problem in the dual model of Avanzi et al. ([2], [1], and [3]). Using the fluctuation theory of spectrally positive L{e}vy processes, we give a short exposition in which we show the optimality of barrier strategies for all such L{e}vy processes. Moreover, we characterize the optimal barrier using the functional inverse of a scale function. We also consider the capital injection problem of [3] and show that its value function has a very similar form to the one in which the horizon is the time of ruin.