On the law of the supremum of Lévy processes

We show that the law of the overall supremum overline{X}_{t}=sup_{slet}X_{s}overline{X}_{t}=sup_{slet}X_{s} of a Lévy process XX, before the deterministic time tt is equivalent to the average occupation measure μ+t(dx)=∫t0P(Xs∈dx)dsmu_{t}^{+}(dx)=int_{0}^{t}mathbb{P} (X_{s}in dx),ds, whenever 0 is regular for both open halflines (−∞,0)(-infty,0) and (0,∞)(0,infty). In this case, P(X¯¯¯t∈dx)mathbb{P} (overline{X}_{t}in dx) is absolutely continuous for some (and hence for all) t>0t>0 if and only if the resolvent measure of XX 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)(g_{t},overline{X}_{t}) and (gt,X¯¯¯t,Xt)(g_{t},overline{X}_{t},X_{t}), where gtg_{t} is the time at which the supremum occurs before tt. The proofs of these results use an expression of the joint law P(gt∈ds,Xt∈dx,X¯¯¯t∈dy)mathbb{P} (g_{t}in ds,X_{t}in dx,overline{X}_{t}in 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