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Nonlinear Lévy processes and their characteristics


Authors: Ariel Neufeld and Marcel Nutz
Journal: Trans. Amer. Math. Soc. 369 (2017), 69-95
MSC (2010): Primary 60G51, 60G44
DOI: https://doi.org/10.1090/tran/6656
Published electronically: March 9, 2016
MathSciNet review: 3557768
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Abstract: We develop a general construction for nonlinear Lévy processes with given characteristics. More precisely, given a set $ \Theta $ of Lévy triplets, we construct a sublinear expectation on Skorohod space under which the canonical process has stationary independent increments and a nonlinear generator corresponding to the supremum of all generators of classical Lévy processes with triplets in $ \Theta $. The nonlinear Lévy process yields a tractable model for Knightian uncertainty about the distribution of jumps for which expectations of Markovian functionals can be calculated by means of a partial integro-differential equation.


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Additional Information

Ariel Neufeld
Affiliation: Department of Mathematics, ETH Zurich, 8092 Zurich, Switzerland
Email: ariel.neufeld@math.ethz.ch

Marcel Nutz
Affiliation: Departments of Statistics and Mathematics, Columbia University, New York, New York 10027
Email: mnutz@columbia.edu

DOI: https://doi.org/10.1090/tran/6656
Keywords: Nonlinear L\'evy process, sublinear expectation, partial integro-differential equation, semimartingale characteristics, Knightian uncertainty
Received by editor(s): January 28, 2014
Received by editor(s) in revised form: November 29, 2014
Published electronically: March 9, 2016
Additional Notes: The first author gratefully acknowledges the financial support of Swiss National Science Foundation Grant PDFMP2-137147/1
The second author gratefully acknowledges the financial support of NSF Grant DMS-1208985
Article copyright: © Copyright 2016 American Mathematical Society