Statistical inference for nonergodic weighted fractional Vasicek models

A problem of drift parameter estimation is studied for a nonergodic weighted fractional Vasicek model defined as dX_t = θ(μ+X_t)dt +dBt^a,b, t ≥ 0, with unknown parameters θ>0, μ ∈ ℝ and α:= θμ, whereas B^a,b:= {Bt^a,b,t ≥ 0} is a weighted fractional Brownian motion with parameters a>−1, |b| < 1, |b| <a+ 1. Least square-type estimators (˜θ_T, ˜μ_T) and (˜θ_T,˜α_T) are provided, respectively, for (θ, μ) and (θ, α) based on a continuous-time observation of {X_t, t ∈[0,T]} as T →∞. The strong consistency and the joint asymptotic distribution of (˜θ_T, ˜μ_T) and (˜θ_T,˜α_T) are studied. Moreover, it is obtained that the limit distribution of ˜θ_T is a Cauchy-type distribution, and ˜μ_T and ˜α_T are asymptotically normal.