Statistical inference for models driven by 𝑛-th order fractional Brownian motion

Abstract

We consider the following stochastic integral equation X ( t ) = μ t + σ ∫ 0 t φ ( s ) d B H n ( s ) X(t)=\mu t + \sigma \int _0^t \varphi (s) dB_H^n(s) , t ≥ 0 t\geq 0 , where φ \varphi is a known function and B H n B^n_H is the n n -th order fractional Brownian motion. We provide explicit maximum likelihood estimators for both μ \mu and σ 2 \sigma ^2 , then we formulate explicitly a least squares estimator for μ \mu and an estimator for σ 2 \sigma ^2 by using power variations method. The consistency and asymptotic normality are established for those estimators when the number of observations or the time horizon is sufficiently large.