Optimal dividend problem with a terminal value for spectrally positive Lévy 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 Lévy process before dividends are deducted. Using the fluctuation theory of spectrally positive Lévy 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).