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

TL;DR

The paper analyzes semilinear SDEs driven by a fractional Brownian motion with Hurst index $H\in(\tfrac{1}{2},1)$ and almost periodic coefficients, introducing $\theta$-almost periodicity to study both existence/uniqueness and statistical inference. It proves the existence and uniqueness of a continuous, uniformly bounded $\theta$-almost periodic (in square mean) solution under a contraction condition, using a fixed-point argument for the integral operator $\Gamma$. The statistical contribution shows that a Skorokhod-integral-based least-squares estimator for the drift parameter is consistent in both periodic and almost periodic regimes, leveraging periodic/AP mean-value results and ergodic theory. The results extend the understanding of stochastic dynamics with structured nonstationarity driven by $d$-dimensional fBm and provide a framework for reliable drift estimation in such systems, with potential applications to fractional Langevin-type models.

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.

Theorems & Definitions (31)

• Definition 2.1
• Proposition 2.2
• Proposition 2.3
• Definition 2.4
• Definition 2.5
• Proposition 2.6
• Definition 2.7
• Theorem 2.8
• Proposition 2.9
• Proposition 2.10