On the law of the supremum of L\'evy processes

We show that the law of the overall supremum $\bar{X}_t=\sup_{s\le t}X_s$ of a L\'evy process $X$ before the deterministic time $t$ is equivalent to the average occupation measure \mu_t(dx)=\int_0^t\p(X_s\in dx)\,ds, whenever 0 is regular for both open halflines $(-\infty,0)$ and $(0,\infty)$. In this case, \p(\bar{X}_t\in 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 one of the halflines $(-\infty,0)$ or $(0,\infty)$. Then we give absolute continuity criterions for the laws of $(\bar{X}_t,X_t)$, $(g_t,\bar{X}_t)$ and $(g_t,\bar{X}_t,X_t)$, 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\in ds,X_t\in dx,\bar{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

Invariance principles for random walks conditioned to stay positive

Let $\{S_n\}$ be a random walk in the domain of attraction of a stable law $\mathcal{Y}$, i.e. there exists a sequence of positive real numbers $(a_n)$ such that $S_n/a_n$ converges in law to $\mathcal{Y}$. Our main result is that the rescaled process $(S_{\lfloor nt\rfloor}/a_n, t\ge 0)$, when conditioned to stay positive, converges in law (in the functional sense) towards the corresponding stable L\'{e}vy process conditioned to stay positive. Under some additional assumptions, we also prove a related invariance principle for the random walk killed at its first entrance in the negative half-line and conditioned to die at zero.Comment: Published in at http://dx.doi.org/10.1214/07-AIHP119 the Annales de l'Institut Henri Poincar\'e - Probabilit\'es et Statistiques (http://www.imstat.org/aihp/) by the Institute of Mathematical Statistics (http://www.imstat.org

The lower envelope of positive self-similar Markov processes

We establish integral tests and laws of the iterated logarithm for the lower envelope of positive self-similar Markov processes at 0 and $+\infty$. Our proofs are based on the Lamperti representation and time reversal arguments. These results extend laws of the iterated logarithm for Bessel processes due to Dvoretsky and Erd\"{o}s, Motoo and Rivero

Shifting processes with cyclically exchangeable increments at random

We propose a path transformation which applied to a cyclically exchangeable increment process conditions its minimum to belong to a given interval. This path transformation is then applied to processes with start and end at zero. It is seen that, under simple conditions, the weak limit as epsilon tends to zero of the process conditioned on remaining above minus epsilon exists and has the law of the Vervaat transformation of the process. We examine the consequences of this path transformation on processes with exchangeable increments, L\'evy bridges, and the Brownian bridge.Comment: 14 pages and 3 figure