It is shown how to construct a successful co-adapted coupling of two copies of an n-dimensional Brownian motion (B1, …, Bn) while simultaneously coupling all corresponding copies of Lévy stochastic areas ∫Bi dBj−∫Bj dBi. 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.
"Coupling all the Lévy stochastic areas of multidimensional Brownian motion." Ann. Probab. 35 (3) 935 - 953, May 2007. https://doi.org/10.1214/009117906000001196