$\epsilon$-Strong Simulation for Multidimensional Stochastic Differential Equations via Rough Path Analysis

TL;DR

This paper introduces a method to simulate multidimensional diffusion processes with a guaranteed uniform error bound using rough path analysis, enabling fully simulatable approximations with controllable accuracy.

Contribution

The authors develop an explicit construction of fully simulatable processes approximating multidimensional diffusions within a user-defined error, leveraging rough path theory for strong simulation guarantees.

Findings

Achieves almost sure uniform approximation within epsilon error

Provides an adaptive scheme for reducing approximation error

Utilizes rough path analysis to control approximation quality

Abstract

Consider a multidimensional diffusion process $X=\{X\left(t\right) :t\in\lbrack0,1]\}$. Let $\varepsilon>0$ be a \textit{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 \[ \sup_{0\leq t\leq1}\left\Vert X_{\varepsilon}\left(t\right) -X\left(t\right) \right\Vert_{\infty}<\varepsilon \] with probability one. Moreover, the user can adaptively choose $\varepsilon^{\prime}\in\left(0,\varepsilon\right) $ so that $X_{\varepsilon^{\prime}}$ (also piecewise constant and fully simulatable) can be constructed conditional on $X_{\varepsilon}$ to ensure an error smaller than $\varepsilon^{\prime}>0$ with probability one. Our construction requires a detailed study of continuity estimates of the Ito map using Lyon's theory of rough paths. We approximate the underlying Brownian motion, jointly with the L\'{e}vy areas with a deterministic $\varepsilon$ error in the underlying rough path metric.