A note on one-sided solutions for optimal stopping problems driven by Lévy processes

All optimal stopping problems with increasing, logconcave and right-continuous reward functions under Lévy processes have been shown to admit one-sided solutions. There is relatively little research attention, however, on constructing an explicit solution of the problem when the reward function is increasing and logconcave. Thus, this paper aims to explore the role of the monotonicity and logconcavity in the characterization of the optimal threshold value, particularly in terms of the ladder height process. In this paper, with additional smoothness assumptions on the reward function, we give an explicit expression for the optimal threshold value provided that the underlying Lévy process drifts to −∞ almost surely. In the particular case where the reward function is (x+)ν=(max{x,0})ν, ν∈(0,∞), we establish a connection indicating that the expression coincides with the positive root of the associated Appell function.