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Weak Reflection Principle for Levy Processes

33 Pages Posted: 18 Jul 2014  

Erhan Bayraktar

University of Michigan at Ann Arbor - Department of Mathematics

Sergey Nadtochiy

University of Michigan at Ann Arbor - Department of Mathematics

Date Written: August 15, 2013

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 L'evy 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.

Keywords: Weak reflection principle, barrier options, static hedging, Levy processes

Suggested Citation

Bayraktar, Erhan and Nadtochiy, Sergey, Weak Reflection Principle for Levy Processes (August 15, 2013). Available at SSRN: https://ssrn.com/abstract=2467257 or http://dx.doi.org/10.2139/ssrn.2467257

Erhan Bayraktar (Contact Author)

University of Michigan at Ann Arbor - Department of Mathematics ( email )

2074 East Hall
530 Church Street
Ann Arbor, MI 48109-1043
United States

Sergey Nadtochiy

University of Michigan at Ann Arbor - Department of Mathematics ( email )

2074 East Hall
530 Church Street
Ann Arbor, MI 48109-1043
United States

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