Coupling time distribution asymptotics for some couplings of the Levy stochastic area

TL;DR

This paper develops explicit couplings for Brownian motion and Levy stochastic areas, deriving asymptotic distributions of coupling times in two dimensions and bounds in higher dimensions, advancing understanding of stochastic coupling behaviors.

Contribution

It introduces explicit co-adapted couplings for Brownian motion and Levy areas, providing exact asymptotics in 2D and bounds in higher dimensions using Dufresne's formula.

Findings

Exact asymptotics for coupling times in 2D

Upper and lower bounds for higher dimensions

Application of Dufresne's formula to stochastic coupling

Abstract

We exhibit some explicit co-adapted couplings for n-dimensional Brownian motion and all its Levy stochastic areas. In the two-dimensional case we show how to derive exact asymptotics for the coupling time under various mixed coupling strategies, using Dufresne's formula for the distribution of exponential functionals of Brownian motion. This yields quantitative asymptotics for the distributions of random times required for certain simultaneous couplings of stochastic area and Brownian motion. The approach also applies to higher dimensions, but will then lead to upper and lower bounds rather than exact asymptotics.