Numerical approximation of SDEs with fractional noise and distributional drift

TL;DR

This paper develops numerical methods for stochastic differential equations driven by fractional Brownian motion with irregular, distributional drifts, providing explicit convergence rates under certain regularity conditions.

Contribution

It extends the analysis of Euler schemes to SDEs with highly singular drifts driven by fractional Brownian motion, including cases with minimal regularity.

Findings

Explicit convergence rate for tamed Euler scheme when drift regularity exceeds a threshold.

Recovery of strong well-posedness for SDEs with distributional drifts.

Introduction of new regularising properties of discrete-time fractional Brownian motion.

Abstract

We study the numerical approximation of SDEs with singular drifts (including distributions) driven by a fractional Brownian motion. Under the Catellier-Gubinelli condition that imposes the regularity of the drift to be strictly greater than $1-1/(2H)$, we obtain an explicit rate of convergence of a tamed Euler scheme towards the SDE, extending results for bounded drifts. Beyond this regime, when the regularity of the drift is $1-1/(2H)$, we derive a non-explicit rate. As a byproduct, strong well-posedness for these equations is recovered. Proofs use new regularising properties of discrete-time fBm and a new critical Gr\"onwall-type lemma. We present examples and simulations.