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.

Weighted Pseudo-θ-Almost Periodic Sequence and Finite-Time Guaranteed Cost Control for Discrete-Space and Discrete-Time Stochastic Genetic Regulatory Networks with Time Delays

“…On the other hand, the global exponential stability and almost periodic nature of GRNs are significant and necessary dynamical behaviours that have been extensively researched by many authors in the last two decades, see the literature [18,[38][39][40][41][42]. Particularly in stochastic models, the notion of θ-almost periodicity was first introduced in the paper [43] on the basis of semi-flow and metric dynamical system theories, and the existence of θalmost periodicity for several continuous-time stochastic models was investigated [44,45]. First, pseudo-almost periodicity was introduced in the early 1990s by Zhang [46] as a natural extension of classical probability periodicity.…”

Nonparametric estimation of the trend in reflected fractional SDE

“…The oldest kind of estimators of the drift function b 0 is based on the long-time behavior of the solution of Equation (1). In the parametric estimation framework, the reader may refer to the monograph [8] written by K. Kubilius, Y. Mishura and K. Ralchenko, but also to Kleptsyna and Le Breton [7], Tudor and Viens [15], Hu and Nualart [5], Neuenkirch and Tindel [12], Hu et al [6], Marie and Raynaud de Fitte [11], etc. In the nonparametric estimation framework, the reader may refer to Saussereau [14] and Comte and Marie [2].…”

“…The stochastic integral involved in the definition of the estimators studied in [5], [6], [11], [2], [3] and [10] is taken in the sense of Skorokhod. To be not directly calculable from one observation of X is the major drawback of the Skorokhod integral with respect to X.…”