We proove a new duality relation between stable Lévy processes with index α ∈ [ 1 2 , 1] and those with index 1 α . This duality appears to be the trajectorial version of the duality of Zolotarev which concerns one dimensional stable laws. We give an application of this result to the behaviour of the paths at small and large times of the process "conditioned to stay positive".

Lewis and Mordecki have computed the Wiener-Hopf factorization of a Lévy process whose restriction on ]0, +∞[ of their Lévy measure has a rational Laplace transform. That allows to compute the distribution of (Xt, inf 0≤s≤t Xs). For the same class of Lévy processes, we compute the distribution of (Xt, inf 0≤s≤t Xs, sup 0≤s≤t Xs) and also the behavior of this triple at certain stopping time, like the first exit time of an interval containing the origin. Some applications to the pricing of double barrier options with or without rebate are evocated.

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We prove a conjecture of J. Bertoin: a Le vy process has increase times if and only if the integral 0 1qx dr x is ®nite, where q and r are the distribution functions of the minimum and the maximum of the Le vy process killed at an independent exponential time. The``if'' part of the statement had been obtained before by R. Doney. Our proof uses dierent techniques, from potential theory and the general theory of processes, and is self-contained. Our results also show that if t b 0 1a2 for all t small enough, then the process does not have increase times.

Abstract. We establish a connection between the inverse scattering problem and the determination of the distribution of the position of a Lévy process at the exit time of a bounded interval in term of its Lévy exponent.