On the law of the supremum of Lévy processes

Abstract

We show that the law of the overall supremum X t = sup s≤t X s of a Lévy process X, before the deterministic time t is equivalent to the average occupation measure μ + t (dx) = t 0 P(X s ∈ dx) ds, whenever 0 is regular for both open halflines (−∞, 0) and (0, ∞). In this case, P(X t ∈ dx) is absolutely continuous for some (and hence for all) t > 0 if and only if the resolvent measure of X is absolutely continuous. We also study the cases where 0 is not regular for both halflines. Then we give absolute continuity criterions for the laws of (g t , X t ) and (g t , X t , X t ), where g t is the time at which the supremum occurs before t. The proofs of these results use an expression of the joint law P(g t ∈ ds, X t ∈ dx, X t ∈ dy) in terms of the entrance law of the excursion measure of the reflected process at the supremum and that of the reflected process at the infimum. As an application, this law is made (partly) explicit in some particular instances.

The Entrance Law of the Excursion Measure of the Reflected Process for Some Classes of Lévy Processes

“…In the light of Theorem 6 in [3], the entrance laws q t (dx) and q * t (dx) seem to be basic objects in the study of the supremum distributions. More precisely, under our assumption that 0 is regular for both negative and positive half-lines, the representation (4.4) from [3] reads as…”

Short proofs in extrema of spectrally one sided Lévy processes

“…Then for λ ≥ 0 and z < 0, define the Laplace transform of the function t → t −1 p t (z) by Recall from [9] and [10] the definition of the entrance laws q t (dx) (resp. q * t (dx)) of the excursions reflected at the supremum (resp.…”

“…Both reflected processes X − X and X − X are homogeneous Markov processes. We denote by n and n * the characteristic measures of the corresponding Poisson point processes of excursions away from 0, see [9]. Then q t (dx) and q * t (dx) are defined by where ζ denotes the life time of the excursions and f is any positive Borel function.…”

Queues and Lévy Fluctuation Theory

“….x/ as introduced in Chapter 2. to [60] or [161] for explicit criteria under which (12.6) holds. 6.4].…”