Quasi-Monte Carlo for SDE Simulation: Error Analysis and Dimensionality Reduction

Abstract

We investigate the numerical simulation of general stochastic differential equations (SDEs) using Quasi-Monte Carlo (QMC) methods. First, we provide a rigorous theoretical analysis of the QMC method applied to the Euler-Maruyama (EM) scheme, establishing that it significantly accelerates the decay of the sampling error and achieves an asymptotically superior convergence rate over the classical Monte Carlo method. Second, the traditional EM scheme exhibits a slow polynomial decay of the discretization error, which necessitates a large number of time steps and leads to a significantly high integration dimension. To address this issue, we propose a Multilevel Stochastic Time Grid (MSTG) method based on Exact Simulation techniques, and we rigorously establish its convergence rate under randomized QMC sampling, proving that it preserves the high-order convergence of the sampling error. In terms of the overall error, the truncation error of the proposed MSTG method exhibits a remarkably fast super-exponential decay. Consequently, to achieve a given accuracy level, our approach requires significantly fewer discretization steps than the EM scheme, thereby drastically reducing the actual integration dimension of the QMC method. This substantial dimensionality reduction strategy greatly enhances the practical efficiency of the QMC algorithm. Numerical experiments fully corroborate the superiority of the proposed approach.