Symmetry and Time Changed Brownian Motions

TL;DR

This paper investigates the symmetry properties of Brownian subordinations with drift, establishing a precise condition under which the process exhibits symmetry, specifically when the drift equals -1/2, in the context of Lévy processes.

Contribution

It provides a necessary and sufficient condition for symmetry in Brownian subordinations with drift, extending previous work to Lévy processes.

Findings

Symmetry holds if and only if drift = -1/2 for Lévy process subordinations.

Characterizes when Brownian subordination with drift exhibits symmetry.

Extends symmetry criteria to Lévy process frameworks.

Abstract

In this paper we examine which Brownian Subordination with drift exhibits the symmetry property introduced by Fajardo and Mordecki (2006). We obtain that when the subordination results in a L\'evy process, a necessary and sufficient condition for the symmetry to hold is that drift must be equal to -1/2.