The maximum of a Lévy process reflected at a general barrier

We investigate the reflection of a Lévy process at a deterministic, time-dependent barrier and in particular properties of the global maximum of the reflected Lévy process. Under the assumption of a finite Laplace exponent, [psi]([theta]), and the existence of a solution [theta]*>0 to [psi]([theta])=0 we derive conditions in terms of the barrier for almost sure finiteness of the maximum. If the maximum is finite almost surely, we show that the tail of its distribution decays like Kexp(-[theta]*x). The constant K can be completely characterized, and we present several possible representations. Some special cases where the constant can be computed explicitly are treated in greater detail, for instance Brownian motion with a linear or a piecewise linear barrier. In the context of queuing and storage models the barrier has an interpretation as a time-dependent maximal capacity. In risk theory the barrier can be interpreted as a time-dependent strategy for (continuous) dividend pay out.