Lévy area of fractional Ornstein-Uhlenbeck process and parameter estimation

In this paper, we study the estimation problem of an unknown drift parameter matrix for fractional Ornstein-Uhlenbeck process in multi-dimensional setting. By using rough path theory, we propose pathwise rough path estimators based on both continuous and discrete observations of a single path. The approach is applicable to the high-frequency data. To formulate the parameter estimators, we define a theory of pathwise It\^o integrals with respect to fractional Brownian motion. By showing the regularity of fractional Ornstein-Uhlenbeck processes and the long time asymptotic behaviour of the associated L\évy area processes, we prove that the estimators are strong consistent and pathwise stable. Numerical studies and simulations are also given in this paper.