We study the well-posedness and numerical approximation of multidimensional stochastic differential equations (SDEs) with distributional drift, driven by a fractional Brownian motion. First, we prove weak existence for such SDEs. This holds under a condition that relates the Hurst parameter $H$ of the noise to the Besov regularity of the drift. Then under a stronger condition, we study the error between a solution $X$ of the SDE with drift $b$ and its tamed Euler scheme with mollified drift $b^n$. We obtain a rate of convergence in $L^m(\Omega)$ for this error, which depends on the Besov regularity of the drift. This result covers the critical case of the regime of strong existence and pathwise uniqueness. When the Besov regularity increases and the drift becomes a bounded measurable function, we recover the (almost) optimal rate of convergence $1/2-\varepsilon$. As a byproduct of this convergence, we deduce that pathwise uniqueness holds in a class of H\"older continuous solutions and that any such solution is strong. The proofs rely on stochastic sewing techniques, especially to deduce new regularising properties of the discrete-time fractional Brownian motion. We also present several examples and numerical simulations that illustrate our results.
翻译:我们研究的是分布式流动的多维随机差异方程式(SDEs)的准确性和数字近似性。 首先,我们证明这种SDEs存在薄弱。 这符合一个条件, 该条件与漂移的贝索夫常规性有关, 其噪音的赫斯特参数$H$与贝索夫正常性有关。 然后, 在一个更强大的条件下, 我们研究SDE以漂移美元为单位的溶液( X$ ) 与其以平滑的漂移美元为单位的软化尤尔格方案之间的误差。 作为这一汇合的副产品, 我们推断出, 以美元(\ 奥梅加) $($) 来计算这一错误的趋同率, 这取决于漂移的贝索夫常规性。 这个结果涵盖了强势存在和路径独特性制度的关键案例。 当Besov的周期性上升和漂移成为一种受约束的可测量功能时, 我们恢复了(最接近的) 最佳趋同率1/2\ varepslonalonal$。 作为这一趋同的副产品, 我们推断出路径的独特性的独特性在一种H\\\\\\\\\\caldeal rodeal rodeal rodeal rodeal rodeal extime extime extical extime exildal ex ex extime ex