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 T_b* (r), at which X, started with X ₀ = 0, first leaves the space-time regions {(t, y) ∈ ℝ²: y ≤ rt^b, t ≥ 0} (one-sided exit), or {(t, y) ∈ ℝ²: |y| ≤ rt^b, t ≥ 0} (two-sided exit), 0 ≤ b < 1, as r ↓ 0. Thus essentially we determine whether or not these passage times behave like deterministic functions in the sense of different modes of convergence; specifically convergence in probability, almost surely and in L^p. In many instances these are seen to be equivalent to relative stability of the process X itself. The analogous large time problem is also discussed.