We study the numerical approximation of multidimensional stochastic differential equations (SDEs) with distributional drift, driven by a fractional Brownian motion. We work under the Catellier-Gubinelli condition for strong well-posedness, which assumes that the regularity of the drift is strictly greater than $1-1/(2H)$, where $H$ is the Hurst parameter of the noise. The focus here is on the case $H<1/2$, allowing the drift $b$ to be a distribution. We compare the solution $X$ of the SDE with drift $b$ and its tamed Euler scheme with mollified drift $b^n$, to obtain an explicit rate of convergence for the strong error. This extends previous results where $b$ was assumed to be a bounded measurable function. In addition, we investigate the limit case when the regularity of the drift is equal to $1-1/(2H)$, and obtain a non-explicit rate of convergence. As a byproduct of this convergence, there exists a strong solution that is pathwise unique in a class of H\"older continuous solutions. The proofs rely on stochastic sewing techniques, especially to deduce new regularising properties of the discrete-time fractional Brownian motion. In the limit case, we introduce a critical Gr\"onwall-type lemma to quantify the error. We also present several examples and numerical simulations that illustrate our results.

翻译：本文研究由分数布朗运动驱动的、具有分布漂移的多维随机微分方程（SDE）的数值逼近方法。我们在Catellier-Gubinelli强适定性条件下开展工作，该条件要求漂移的正则性严格大于$1-1/(2H)$（其中$H$为噪声的Hurst参数）。本文重点关注$H<1/2$的情形，此时允许漂移$b$为分布函数。通过比较带漂移$b$的SDE解$X$与采用磨光漂移$b^n$的驯化欧拉格式，我们获得了强误差的显式收敛速率。该结果扩展了此前将$b$假设为有界可测函数的研究。此外，我们考察了漂移正则性等于$1-1/(2H)$的极限情形，并得到非显式收敛速率。作为该收敛性的副产品，我们获得了一类Hölder连续解中具有路径唯一性的强解。证明过程采用随机缝合技术，尤其用于推导离散时间分数布朗运动的新正则化性质。在极限情形中，我们引入临界型Grönwall引理以量化误差。本文还提供多个数值算例和模拟结果以验证理论分析。