Regularization by regular noise: a numerical result

TL;DR

The paper analyzes the Euler–Maruyama discretization of the singular SDE $dX_t=b(X_t)\,dt+dB_t^H$ with $H>1$ and $b\in C^\alpha$, $\alpha>1-\frac{1}{2H}$. It establishes a strong convergence rate of $n^{-1}$ for $X^n$ to the unique solution $X$, using the stochastic sewing lemma and Gaussian regularization from the fractional Brownian noise to circumvent Girsanov limitations in the non-Markovian setting. Under the stronger assumption $b\in C^1$, it proves that $n(X-X^n)$ converges to a non-trivial limit, thereby confirming the rate $n^{-1}$ is optimal for the scheme. The work also situates these results within the broader context of regularization by noise and extends numerical analysis techniques for non-Markovian, singular drift SDEs. Collectively, the results provide a precise quantitative understanding of discretization error for a class of SDEs driven by regular noise with Hurst index $H>1$.

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

Theorems & Definitions (18)

• Theorem 2.2
• Remark 2.3
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3
• proof
• Lemma 4.1
• proof
• Lemma 4.2
• proof