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We analyze the optimal dividend payment problem in the dual model under constant transaction costs. We show, for a general spectrally positive Lévy process, an optimal strategy is given by a (c1,c2)(c1,c2)-policy that brings the surplus process down to c1c1 whenever it reaches or exceeds c2c2 for some 0≤c1<c20≤c1<c2. 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).

We solve the optimal dividend problem under fixed transaction costs in the so-called dual model, in which the surplus of a company is driven by a Lévy process with positive jumps (spectrally positive Lévy process). This is an appropriate model for a company driven by inventions or discoveries. The case without transaction costs has recently been well-studied; see Avanzi et al. (2007), Bayraktar and Egami (2008), Avanzi and Gerber (2008), and Avanzi et al. (2011). In particular, in Bayraktar et al. (2013), we show the optimality of a barrier strategy (reflected Lévy process) for a general spectrally positive Lévy process of bounded or unbounded variation.