ON THE LAW OF THE SUPREMUM OF LÉVY PROCESSES

• Published in: The Annals of probability 2013-05, Vol.41 (3), p.1191-1217
• Main Author: Chaumont, L.
• Format: Article
• Language: English
• Subjects: 60G51; absolute continuity; average occupation measure; Borel sets; Density; equivalent measures; excursion measure; Fluctuation theorem; Laplace transformation; Lebesgue measures; Mathematical analysis; Mathematics; Past supremum; Poisson process; Positive laws; Probability distribution; Proof theory; Random variables; reflected process; Semigroups; Set theory; Studies
• ISSN: 0091-1798; 2168-894X
• DOI: 10.1214/11-aop708

We show that the law of the overall supremum X̄ t = sup s≤t X s of a Lévy process X, before the deterministic time t is equivalent to the average occupation measure ${\mathrm{\mu }}_{\mathrm{t}}^{+}\left(\mathrm{d}\mathrm{x}\right)={\int }_{0}^{\mathrm{t}}\mathrm{\mathbb{P}}({\mathrm{X}}_{\mathrm{s}}\in \mathrm{d}\mathrm{x}) \ \mathrm{d}\mathrm{s}$ , whenever 0 is regular for both open halflines (-∞, 0) and (0, ∞). In this case, ℙ(X̄ 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 , X̄ t ) and (g t , X̄ 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, X̄ 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.