Faking Brownian motion with continuous Markov martingales

TL;DR

This paper constructs continuous Markov martingales with Brownian marginals that are not true Brownian motions, challenging the uniqueness of Brownian motion as the only such process with these properties.

Contribution

It introduces a novel class of 'very fake' Brownian motions, continuous Markov martingales with Brownian marginals that lack the strong Markov property.

Findings

Constructed examples of continuous Markov martingales with Brownian marginals.

Demonstrated these processes are not true Brownian motions due to absence of strong Markov property.

Challenged the uniqueness of Brownian motion among continuous strong Markov martingales with Brownian marginals.

Abstract

Hamza-Klebaner posed the problem of constructing martingales with Brownian marginals that differ from Brownian motion, so called fake Brownian motions. Besides its theoretical appeal, the problem represents the quintessential version of the ubiquitous fitting problem in mathematical finance where the task is to construct martingales that satisfy marginal constraints imposed by market data. Non-continuous solutions to this challenge were given by Madan-Yor, Hamza-Klebaner, Hobson, and Fan-Hamza-Klebaner whereas continuous (but non-Markovian) fake Brownian motions were constructed by Oleszkiewicz, Albin, Baker-Donati-Yor, Hobson, Jourdain-Zhou. In contrast it is known from Gy\"ongy, Dupire, and ultimately Lowther that Brownian motion is the unique continuous strong Markov martingale with Brownian marginals. We took this as a challenge to construct examples of a "very fake'' Brownian motion, that is, continuous Markov martingales with Brownian marginals that miss out only on the strong Markov property.