Weak reflection principle for L\'evy processes

TL;DR

This paper introduces a weak reflection principle for Lévy processes, extending classical methods to asymmetric processes and enabling new solutions in finance, computation, and inverse problems.

Contribution

It develops a new weak reflection principle applicable to a broader class of stochastic processes, including Lévy processes with one-sided jumps, beyond symmetric diffusions.

Findings

The weak reflection principle applies to various time-homogeneous diffusions.

Extension of the method to Lévy processes with one-sided jumps.

Applications demonstrated in finance, computational methods, and inverse problems.

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\'{e}vy 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.