Small and large time stability of the time taken for a Lévy process to cross curved boundaries

This paper is concerned with the small time behaviour of a Lévy process X. In particular, we investigate the stabilities of the times, T̄ b(r) and Tb* (r), at which X, started with X 0 = 0, first leaves the space-time regions {(t, y) ∈ ℝ2: y ≤ rt