On the pathwise approximation of stochastic differential equations

TL;DR

This paper develops a pathwise convergence analysis for one-step methods solving stochastic differential equations, applying rough path theory to Euler-Maruyama with adaptive strategies, and demonstrating convergence through computational experiments.

Contribution

It introduces a novel pathwise convergence proof for SDE numerical methods using rough path theory, independent of pth mean convergence, and develops an adaptive Euler-Maruyama method.

Findings

The adaptive Euler-Maruyama method converges under the proposed framework.

The theory provides an error-control strategy for SDE integration.

Computational experiments validate the adaptive method's effectiveness.

Abstract

We consider one-step methods for integrating stochastic differential equations and prove pathwise convergence using ideas from rough path theory. In contrast to alternative theories of pathwise convergence, no knowledge is required of convergence in pth mean and the analysis starts from a pathwise bound on the sum of the truncation errors. We show how the theory is applied to the Euler-Maruyama method with fixed and adaptive time-stepping strategies. The assumption on the truncation errors suggests an error-control strategy and we implement this as an adaptive time-stepping Euler-Maruyama method using bounded diffusions. We prove the adaptive method converges and show some computational experiments.