Almost Periodic and Periodic Solutions of Differential Equations Driven by the Fractional Brownian Motion with Statistical Application

Abstract

We show that the unique solution to a semilinear stochastic differential equation with almost periodic coefficients driven by a fractional Brownian motion is almost periodic in a sense related to random dynamical systems. This type of almost periodicity allows for the construction of a consistent estimator of the drift parameter in the almost periodic and periodic cases.

Nonparametric Estimation for I.I.D. Paths of Fractional SDE

“…In [30], Hu, Nualart and Zhou extend this estimator to equations with a drift function depending linearly on the unknown parameter. Finally, in [36], Marie and Raynaud de Fitte extend this estimator to non-homogeneous semi-linear equations with almost periodic coefficients. Now, considering discrete time observations, still in parametric context, Tindel and Neuenkirch [39] study a least squares-type estimator defined by an objective function, tailor-maid with respect to the main result of Tudor and Viens [50] on the rate of convergence of the quadratic variation of the fractional Brownian motion.…”