ε-strong simulation for multidimensional stochastic differential equations via rough path analysis

Consider a multidimensional diffusion process X = {X(t) : t ∈ [0, 1]}. Let ε > 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_ε such that sup ∥X_ε(t) - X(t)∥_∞ < ε 0≤t≤1 with probability one. Moreover, the user can adaptively choose ε′ ∈ (0, ε) so that X_ε′ (also piecewise constant and fully simulatable) can be constructed conditional on X_ε to ensure an error smaller than ε′ with probability one. Our construction requires a detailed study of continuity estimates of the Itô map using Lyons' theory of rough paths. We approximate the underlying Brownian motion, jointly with the Lévy areas with a deterministic ε error in the underlying rough path metric.