L\'evy Area Analysis and Parameter Estimation for fOU Processes via Non-Geometric Rough Path Theory

Abstract

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.