, Volume 22, Issue 2, pp 418-440

On an Explicit Skorokhod Embedding for Spectrally Negative Lévy Processes

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Abstract

We present an explicit solution to the Skorokhod embedding problem for spectrally negative Lévy processes. Given a process X and a target measure μ satisfying an explicit admissibility condition we define functions φ ± such that the stopping time T=inf {t>0:X t ∈{−φ (L t ),φ +(L t )}} induces X T μ, where (L t ) is the local time in zero of X. We also treat versions of T which take into account the sign of the excursion straddling time t. We prove that our stopping times are minimal and we describe criteria under which they are integrable. We compare our solution with the one proposed by Bertoin and Le Jan (Ann. Probab. 20(1):538–548, [1992]). In particular, we compute explicitly the quantities introduced in Bertoin and Le Jan (Ann. Probab. 20(1):538–548, [1992]) in our setup.

Our method relies on some new explicit calculations relating scale functions and the Itô excursion measure of X. More precisely, we compute the joint law of the maximum and minimum of an excursion away from 0 in terms of the scale function.

Research of J. Obłój supported by a Marie Curie Intra-European Fellowship within the 6th European Community Framework Programme.
Research of M. Pistorius supported by EPSRC grant EP/D039053/1.