The Annals of Applied Probability

Weak reflection principle for Lévy processes

Erhan Bayraktar and Sergey Nadtochiy

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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.

Article information

Source
Ann. Appl. Probab. Volume 25, Number 6 (2015), 3251-3294.

Dates
Received: August 2013
Revised: July 2014
First available in Project Euclid: 1 October 2015

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1443703774

Digital Object Identifier
doi:10.1214/14-AAP1073

Mathematical Reviews number (MathSciNet)
MR3404636

Subjects
Primary: 45Q05: Inverse problems 60J75: Jump processes 91G20: Derivative securities

Keywords
Reflection principle Lévy processes static hedging barrier options

Citation

Bayraktar, Erhan; Nadtochiy, Sergey. Weak reflection principle for Lévy processes. Ann. Appl. Probab. 25 (2015), no. 6, 3251--3294. doi:10.1214/14-AAP1073. https://projecteuclid.org/euclid.aoap/1443703774.


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