Approximation of SDEs -- a stochastic sewing approach

TL;DR

This paper introduces a new stochastic sewing approach to analyze the error of SDE approximations, demonstrating convergence of Euler-Maruyama schemes for fractional Brownian motion driven SDEs with non-regular drift and establishing near-optimal rates.

Contribution

It develops a novel stochastic sewing lemma-based method for error analysis, enabling convergence proofs for Euler schemes under minimal regularity assumptions.

Findings

Proves convergence of Euler-Maruyama for fractional Brownian SDEs with non-regular drift.

Establishes strong $L_p$ and almost sure convergence rates of $(1/2+ ext{drift regularity} imes H) ext{ or }1$.

Derives near-optimal convergence rate of $1/2- ext{small epsilon}$ for Brownian-driven SDEs with irregular drift.

Abstract

We give a new take on the error analysis of approximations of stochastic differential equations (SDEs), utilizing and developing the stochastic sewing lemma of L\^e (2020). This approach allows one to exploit regularization by noise effects in obtaining convergence rates. In our first application we show convergence (to our knowledge for the first time) of the Euler-Maruyama scheme for SDEs driven by fractional Brownian motions with non-regular drift. When the Hurst parameter is $H\in(0,1)$ and the drift is $\mathcal{C}^\alpha$, $\alpha\in[0,1]$ and $\alpha>1-1/(2H)$, we show the strong $L_p$ and almost sure rates of convergence to be $((1/2+\alpha H)\wedge 1) -\varepsilon$, for any $\varepsilon>0$. Our conditions on the regularity of the drift are optimal in the sense that they coincide with the conditions needed for the strong uniqueness of solutions from Catellier, Gubinelli (2016). In a second application we consider the approximation of SDEs driven by multiplicative standard Brownian noise where we derive the almost optimal rate of convergence $1/2-\varepsilon$ of the Euler-Maruyama scheme for $\mathcal{C}^\alpha$ drift, for any $\varepsilon,\alpha>0$.