Regularization by regular noise: a numerical result

TL;DR

This paper provides a numerical analysis of a singular stochastic equation driven by fractional Brownian noise, establishing strong convergence of the Euler-Maruyama scheme at rate n^{-1} and confirming its optimality.

Contribution

It proves the strong convergence rate of the Euler-Maruyama approximation for a regularized stochastic equation driven by fractional Brownian noise.

Findings

Euler-Maruyama scheme converges with rate n^{-1}

n(X - X^n) converges to a non-trivial limit under additional smoothness

Rate n^{-1} is confirmed as optimal upper bound

Abstract

We study a singular stochastic equation driven by a regular noise of fractional Brownian type with Hurst index $H \in (1,\infty)\setminus\mathbb{Z}$ and drift coefficient $b \in \mathcal{C}^\alpha$, where $\alpha > 1 - \frac{1}{2H}$. The strong well-posedness of this equation was first established in [Ger23], a phenomenon referred to as regularization by regular noise. In this note, we provide a corresponding numerical analysis. Specifically, we show that the Euler-Maruyama approximation $X^n$ converges strongly to the unique solution $X$ with rate $n^{-1}$. Furthermore, under the additional assumption $b \in \mathcal{C}^1$, we show that $n(X - X^n)$ converges to a non-trivial limit as $n \to \infty$, thereby confirming that the rate $n^{-1}$ is in fact optimal upper bound for this scheme.