On an Explicit Skorokhod Embedding for Spectrally Negative Lévy Processes

Jan Obłój () and Martijn Pistorius ()
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Jan Obłój: Imperial College London
Martijn Pistorius: King’s College London

Journal of Theoretical Probability, 2009, vol. 22, issue 2, 418-440

Abstract: 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.

Keywords: Skorokhod embedding; Levy process; Excursion theory; Scale function; Minimal stopping time; 60G44; 60G40; 60G51 (search for similar items in EconPapers)
Date: 2009
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DOI: 10.1007/s10959-008-0157-7

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