Exit problems for spectrally negative Levy processes reflected at either the supremum or the infimum

For a spectrally negative Levy process X on the real line, let S denote its supremum process and let I denote its infimum process. For a > 0, let tau (a) and K (a) denote the times when the reflected processes (Y) over cap := S - X and Y := X - I first exit level a, respectively; let tau_(a) and K_(a) denote the times when X first reaches S-tau(a) and I-k(a), respectively. The main results of this paper concern the distributions of (tau (a), S-tau(a), tau-(a), (Y) over cap (tau(a))) and of (K(a), I-k(a), k_(a)). They generalize some recent results on spectrally negative Levy processes. Our approach relies on results concerning the solution to the two-sided exit problem for X. Such an approach is also adapted to study the excursions for the reflected processes. More explicit expressions are obtained when X is either a Brownian motion with drift or a completely asymmetric stable process.