This paper addresses the estimation problem of an unknown drift parameter matrix for a fractional Ornstein-Uhlenbeck process in a multi-dimensional setting. To tackle this problem, we propose a novel approach based on rough path theory that allows us to construct pathwise rough path estimators from both continuous and discrete observations of a single path. Our approach is particularly suitable for high-frequency data. To formulate the parameter estimators, we introduce a theory of pathwise It\^o integrals with respect to fractional Brownian motion. By establishing the regularity of fractional Ornstein-Uhlenbeck processes and analyzing the long-term behavior of the associated L\'evy area processes, we demonstrate that our estimators are strongly consistent and pathwise stable. Our findings offer a new perspective on estimating the drift parameter matrix for fractional Ornstein-Uhlenbeck processes in multi-dimensional settings, and may have practical implications for fields including finance, economics, and engineering.

翻译：本文研究了多维分数阶Ornstein-Uhlenbeck过程中未知漂移参数矩阵的估计问题。为解决该问题，我们提出了一种基于粗糙路径理论的新方法，该方法允许我们从单一路径的连续或离散观测中构造路径式粗糙路径估计量。我们的方法尤其适用于高频数据。为构建参数估计量，我们引入了关于分数阶布朗运动的路径式Itô积分理论。通过建立分数阶Ornstein-Uhlenbeck过程的正则性，并分析相关Lévy面积过程的长期行为，我们证明了所提估计量具有强一致性及路径稳定性。本研究为多维分数阶Ornstein-Uhlenbeck过程的漂移参数矩阵估计提供了新视角，并可能对金融、经济学及工程等领域产生实际应用价值。