Optimal dividend problem with a terminal value for spectrally positive Levy processes

In this paper we consider a modified version of the classical optimal dividend problem taking into account both expected dividends and the time value of ruin. We assume that the risk process is modeled by a general spectrally positive Levy process before dividends are deducted. Using the fluctuation theory of spectrally positive Levy processes we give an explicit expression of the value function of a barrier strategy. Subsequently we show that a barrier strategy is the optimal strategy among all admissible ones. Our work is motivated by the recent work of Bayraktar, Kyprianou and Yamazaki (2013a). (C) 2013 The Authors. Published by Elsevier B.V. All rights reserved.