$\varepsilon$-Strong simulation for multidimensional stochastic differential equations via rough path analysis

Abstract

Consider a multidimensional diffusion process $X=\{X(t):t\in [0,1]\}$. Let $\varepsilon>0$ 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 $X_{\varepsilon}$ such that

\[\sup_{0\leq t\leq1}\Vert X_{\varepsilon}(t)-X(t)\Vert_{\infty}<\varepsilon\] with probability one. Moreover, the user can adaptively choose $\varepsilon^{\prime}\in (0,\varepsilon )$ so that $X_{\varepsilon^{\prime}}$ (also piecewise constant and fully simulatable) can be constructed conditional on $X_{\varepsilon}$ to ensure an error smaller than $\varepsilon^{\prime}$ 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 $\varepsilon$ error in the underlying rough path metric.