Minimum L ₁ -norm estimation for fractional Ornstein-Uhlenbeck type process driven by a Hermite process

We investigate the problem of estimation of the drift parameter θ of the fractional Ornstein-Uhlenbeck process dXt=θXtdt+ϵdZtq,H,X0=x0,0≤t≤1, where the process {Ztq,H,0≤t≤1} is a Hermite process and ϵ is a small noise. We study the asymptotic properties of the minimum L1-norm estimator θϵ∗=arg⁡minθ∫01|Xt−xt(θ)|dt as ϵ→0. Here xt(θ)=x0exp⁡{θt}. We prove that the estimator is consistent and that the random variable ϵ−1(θϵ∗−θ) has a limiting distribution as ϵ→0.