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

Consider a multidimensional diffusion process X = {X(t) : t . [0,1]}. Let . > 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. such that $\begin{array}{*{20}{c}} {\sup } \\ {0 \leqslant t \leqslant 1} \\ \end{array} {\left\| {{X_\varepsilon }(t) - X(t)} \right\|_\infty } < \varepsilon $ with probability one. Moreover, the user can adaptively choose .' . (0, .) so that X.' (also piecewise constant and fully simulatable) can be constructed conditional on X. to ensure an error smaller than .' 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 . error in the underlying rough path metric.