Pathwise approximation of SDEs by coupling piecewise abelian rough paths

Abstract

We present a new pathwise approximation scheme for stochastic differential equations driven by multidimensional Brownian motion which does not require the simulation of L\'{e}vy area and has a Wasserstein convergence rate better than the Euler scheme's strong error rate of $O(\sqrt{h})$, where $h$ is the step-size. By using rough path theory we avoid imposing any non-degenerate H\"{o}rmander or ellipticity assumptions on the vector fields of the SDE, in contrast to the similar papers of Alfonsi, Davie, and Malliavin et al. The scheme is based on the log-ODE method with the L\'{e}vy area increments replaced by Gaussian approximations with the same covariance structure. The Wasserstein coupling is achieved by making small changes to the argument of Davie, the latter being an extension of the Koml\'{o}s-Major-Tusn\'{a}dy Theorem. We prove that the convergence of the scheme in the Wasserstein metric is of the order $O(h^{1-2/\gamma-\varepsilon})$ when the vector fields are $\gamma$-Lipschitz in the sense of Stein.