Epsilon-Strong Simulation For Multidimensional Stochastic Differential Equations Via Rough Path Analysis

Consider a multidimensional diffusion process X = {X (t) : t is an element of [0, 1]}. Let epsilon > 0 be a deterministic, user defined, tolerance error parameter. Under standard regularity conditions on the drift and diffusion coefficients of X, we construct a probability space, supporting both X and an explicit, piecewise constant, fully simulatable process X-epsilon such that(sup)(0 <= t <= 1)parallel to X-epsilon (t) - X parallel to(infinity) < epsilonwith probability one. Moreover, the user can adaptively choose epsilon' is an element of (0, epsilon) so that X-epsilon' (also piecewise constant and fully simulatable) can be constructed conditional on X-epsilon to ensure an error smaller than epsilon' with probability one. Our construction requires a detailed study of continuity estimates of the Ito map using Lyons' theory of rough paths. We approximate the underlying Brownian motion, jointly with the Levy areas with a deterministic epsilon error in the underlying rough path metric.