In this paper, we prove the existence of strong solutions to an stochastic differential equation with a generalized drift driven by a multidimensional fractional Brownian motion for small Hurst parameters \(H<\frac{1}{2}.\) Here, the generalized drift is given as the local time of the unknown solution process, which can be considered an extension of the concept of a skew Brownian motion to the case of fractional Brownian motion. Our approach for the construction of strong solutions is new and relies on techniques from Malliavin calculus combined with a “local time variational calculus” argument.

中文翻译：

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具有广义漂移和多维分数布朗初始噪声的随机微分方程的强解。

在本文中，我们证明了对于较小的Hurst参数\（H <\ frac {1} {2}。\），具有多维多维布朗运动驱动的具有广义漂移的随机微分方程的强解的存在。漂移以未知解过程的本地时间给出，可以认为是偏布朗运动概念到分数布朗运动情况的扩展。我们构建强解决方案的方法是新的，并且依赖于Malliavin微积分的技术结合“本地时间变分微积分”的论点。