Regularization by regular noise: a numerical result

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 $b \in \mathcal{C}<sup>\alpha$, where $\alpha &gt; 1 - \frac{1}{2H}$. 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 $X<sup>n$ converges strongly to the unique solution $X$ with rate $n<sup>{-1}$. Furthermore, under the additional assumption $b \in \mathcal{C}<sup>1$, we show that $n(X - X<sup>n)$ converges to a non-trivial limit as $n \to \infty$, thereby confirming that the rate $n<sup>{-1}$ is in fact optimal upper bound for this scheme.