NDLI: On an Explicit Skorokhod Embedding for Spectrally Negative Lévy Processes

Content Provider	Springer Nature Link
Author	Obłój, Jan Pistorius, Martijn
Copyright Year	2008
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.
Ending Page	440
Page Count	23
Starting Page	418
File Format	PDF
ISSN	08949840
e-ISSN	15729230
Journal	Journal of Theoretical Probability
Issue Number	2
Volume Number	22
Language	English
Publisher	Springer US
Publisher Date	2008-04-10
Publisher Place	Boston
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Skorokhod embedding Minimal stopping time Levy process Probability Theory and Stochastic Processes Excursion theory Stopping times; optimal stopping problems; gambling theory Scale function Martingales with continuous parameter Statistics Processes with independent increments; LÃ©vy processes
Content Type	Text
Resource Type	Article
Subject	Statistics and Probability Statistics, Probability and Uncertainty

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