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Penalisation of a stable Lévy process involving its one-sided supremum

Kouji Yano, Yuko Yano, Marc Yor (2010)

Annales de l'I.H.P. Probabilités et statistiques

Penalisation involving the one-sided supremum for a stable Lévy process with index α∈(0, 2] is studied. We introduce the analogue of Azéma–Yor martingales for a stable Lévy process and give the law of the overall supremum under the penalised measure.

Potential theory of one-dimensional geometric stable processes

Tomasz Grzywny, Michał Ryznar (2012)

Colloquium Mathematicae

The purpose of this paper is to find optimal estimates for the Green function and the Poisson kernel for a half-line and intervals of the geometric stable process with parameter α ∈ (0,2]. This process has an infinitesimal generator of the form - l o g ( 1 + ( - Δ ) α / 2 ) . As an application we prove the global scale invariant Harnack inequality as well as the boundary Harnack principle.

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