On the law of the supremum of Lévy processes
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L. Chaumont [1]
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[1] University of Angers
University of Angers
Arrondissement d’Angers, Francia
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- Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 41, Nº. 3, 1, 2013, págs. 1191-1217
- Idioma: inglés
- DOI: 10.1214/11-AOP708
- Texto completo no disponible (Saber más ...)
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Resumen
We show that the law of the overall supremum $,$\overline{X}_{t}=\sup_{s\let}X_{s}undefined of a Lévy process X, before the deterministic time t is equivalent to the average occupation measure μ+t(dx)=∫t0P(Xs∈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 (gt,X¯¯¯¯t) and (gt,X¯¯¯¯t,Xt), where gt is the time at which the supremum occurs before t. The proofs of these results use an expression of the joint law P(gt∈ds,Xt∈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.
