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 $b\in {\mathcal{C}}^{\hat{I}}\pm$, where $\hat{I}\pm >1-\frac{1}{2H}$. 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.