Mirror and Synchronous Couplings of Geometric Brownian Motions

TL;DR

This paper investigates the optimality of mirror and synchronous couplings for geometric Brownian motions, revealing their limitations and conditions under which they are optimal or suboptimal across different time horizons.

Contribution

It characterizes the optimality of classical couplings for geometric Brownian motions over finite, ergodic, and infinite horizons, highlighting differences from standard Brownian motion.

Findings

Mirror and synchronous couplings are not always optimal for geometric Brownian motions.

Optimality depends on the time horizon and whether the process is a martingale.

The couplings are always optimal for ergodic average and infinite horizon criteria.

Abstract

The paper studies the question of whether the classical mirror and synchronous couplings of two Brownian motions minimise and maximise, respectively, the coupling time of the corresponding geometric Brownian motions. We establish a characterisation of the optimality of the two couplings over any finite time horizon and show that, unlike in the case of Brownian motion, the optimality fails in general even if the geometric Brownian motions are martingales. On the other hand, we prove that in the cases of the ergodic average and the infinite time horizon criteria, the mirror coupling and the synchronous coupling are always optimal for general (possibly non-martingale) geometric Brownian motions. We show that the two couplings are efficient if and only if they are optimal over a finite time horizon and give a conjectural answer for the efficient couplings when they are suboptimal.