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December 2015 Weak reflection principle for Lévy processes
Erhan Bayraktar, Sergey Nadtochiy
Ann. Appl. Probab. 25(6): 3251-3294 (December 2015). DOI: 10.1214/14-AAP1073

Abstract

In this paper, we develop a new mathematical technique which allows us to express the joint distribution of a Markov process and its running maximum (or minimum) through the marginal distribution of the process itself. This technique is an extension of the classical reflection principle for Brownian motion, and it is obtained by weakening the assumptions of symmetry required for the classical reflection principle to work. We call this method a weak reflection principle and show that it provides solutions to many problems for which the classical reflection principle is typically used. In addition, unlike the classical reflection principle, the new method works for a much larger class of stochastic processes which, in particular, do not possess any strong symmetries. Here, we review the existing results which establish the weak reflection principle for a large class of time-homogeneous diffusions on a real line and then proceed to extend this method to the Lévy processes with one-sided jumps (subject to some admissibility conditions). Finally, we demonstrate the applications of the weak reflection principle in financial mathematics, computational methods and inverse problems.

Citation

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Erhan Bayraktar. Sergey Nadtochiy. "Weak reflection principle for Lévy processes." Ann. Appl. Probab. 25 (6) 3251 - 3294, December 2015. https://doi.org/10.1214/14-AAP1073

Information

Received: 1 August 2013; Revised: 1 July 2014; Published: December 2015
First available in Project Euclid: 1 October 2015

zbMATH: 1330.60064
MathSciNet: MR3404636
Digital Object Identifier: 10.1214/14-AAP1073

Subjects:
Primary: 45Q05, 60J75, 91G20

Rights: Copyright © 2015 Institute of Mathematical Statistics

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Vol.25 • No. 6 • December 2015
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