Coupling iterated kolmogorov diffusions

Abstract

The Kolmogorov (1934) diffusion is the two-dimensional diffusion generated by real Brownian motion B and its time integral ∫ B d t. In this paper we construct successful co-adapted couplings for iterated Kolmogorov diffusions defined by adding iterated time integrals ∫ ∫ B d s d t, … as further components to the original Kolmogorov diffusion. A Laplace-transform argument shows it is not possible successfully to couple all iterated time integrals at once; however we give an explicit construction of a successful co-adapted coupling method for (B; ∫ B d t; ∫ ∫ B d s d t); and a more implicit construction of a successful co-adapted coupling method which works for finite sets of iterated time integrals. © 2004 Applied Probability Trust.