Coupling all the Lévy stochastic areas of multidimensional Brownian motion

Abstract

It is shown how to construct a successful co-adapted coupling of two copies of an n-dimensional Brownian motion (B₁, …, B_n) while simultaneously coupling all corresponding copies of Lévy stochastic areas ∫B_i dB_j−∫B_j dB_i. It is conjectured that successful co-adapted couplings still exist when the Lévy stochastic areas are replaced by a finite set of multiply iterated path- and time-integrals, subject to algebraic compatibility of the initial conditions.