ON THE LAW OF THE SUPREMUM OF LEVY PROCESSES

• 成果类型: Article
• 署名作者: Chaumont, L.
• 署名单位: Universite d'Angers
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/11-AOP708
• 发表日期: 2013
• 页码: 1191-1217
• 关键词: fluctuation identity

摘要

We show that the law of the overall supremum (X) over bar (t) = sup(s <= t) X-s of a Levy process X, before the deterministic time t is equivalent to the average occupation measure mu(+)(t) (dx) = integral(t)(0))P(X-s is an element of dx) ds, whenever 0 is regular for both open halflines (-infinity, 0) and (0, infinity). In this case, P((X) over bar (t) is an element of 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) over bar (t)) and (g(t), (X) over bar (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) is an element of ds, X-t is an element of dx, (X) over bar (t) is an element of 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.