From rough path estimates to multilevel Monte Carlo

TL;DR

This paper explores numerical methods for stochastic differential equations driven by Gaussian noise using rough path theory, emphasizing multilevel Monte Carlo techniques to improve computational efficiency.

Contribution

It introduces a novel analysis combining rough path estimates, Gaussian concentration, and multilevel methods for efficient numerical solutions.

Findings

Multilevel Monte Carlo significantly reduces computational cost.

Numerical examples confirm theoretical efficiency gains.

Abstract

New classes of stochastic differential equations can now be studied using rough path theory (e.g. Lyons et al. [LCL07] or Friz--Hairer [FH14]). In this paper we investigate, from a numerical analysis point of view, stochastic differential equations driven by Gaussian noise in the aforementioned sense. Our focus lies on numerical implementations, and more specifically on the saving possible via multilevel methods. Our analysis relies on a subtle combination of pathwise estimates, Gaussian concentration, and multilevel ideas. Numerical examples are given which both illustrate and confirm our findings.