Optimal periodic–classical barrier strategies for spectrally negative Lévy processes

Abstract We study a stochastic control problem where the underlying process follows a spectrally negative Lévy process. A controller can continuously increase the process but only decrease it at independent Poisson arrival times. We show the optimality of the periodic–classical barrier strategy, which increases the process whenever it would fall below some lower barrier and decreases it whenever it is observed above a higher barrier. An optimal strategy and the value function are written semi-explicitly using scale functions. Numerical results are also given.