Reflection principle and Ocone martingales

TL;DR

This paper proves that a continuous process satisfying reflection symmetry at a sequence of levels converging to zero must be an Ocone local martingale, and explores related invariance properties and open questions.

Contribution

It establishes a new characterization of Ocone martingales based on reflection properties at levels approaching zero, and connects this to invariance and triviality of invariant sigma-fields.

Findings

Processes with reflection symmetry at levels a_n are Ocone martingales.

The triviality of invariant sigma-fields relates to reflection properties.

Results extend to discrete and continuous processes via discretization.

Abstract

Let $M =(M_t)_{t\geq 0}$ be any continuous real-valued stochastic process. We prove that if there exists a sequence $(a_n)_{n\geq 1}$ of real numbers which converges to 0 and such that $M$ satisfies the reflection property at all levels $a_n$ and $2a_n$ with $n\geq 1$, then $M$ is an Ocone local martingale with respect to its natural filtration. We state the subsequent open question: is this result still true when the property only holds at levels $a_n$? Then we prove that the later question is equivalent to the fact that for Brownian motion, the $\sigma$-field of the invariant events by all reflections at levels $a_n$, $n\ge1$ is trivial. We establish similar results for skip free $\mathbb{Z}$-valued processes and use them for the proof in continuous time, via a discretisation in space.