LEAST SQUARES ESTIMATORS OF DRIFT PARAMETER FOR DISCRETELY OBSERVED FRACTIONAL VASICEK-TYPE MODEL

We study the drift parameter estimation problem for a fractional Vasicek-type model X: = {Xt ,t ⩾ 0}, that is defined as dXt = θ(µ + Xt )dt + dBt H, t ⩾ 0 with unknown parameters θ>0 and µ ∈ℝ, where {Bt H,t ⩾ 0} is a fractional Brownian motion of Hurst index H ∈]0, 1[. Let θt ̂and µt ̂ be the least squares-type estimators of θand μ, respectively, based on continuous observation of X. In this paper we assume that the process {Xt ,t ⩾ 0}is observed at discrete time instants ti=iΔn, i=1,n. We analyze discrete versions 0nand µ̃n for θt ̂and µt ̂ respectively. We show that the sequence √nΔn (θn − θ) is tight and √nΔn(μ̃n − μ) is not tight. Moreover, we prove the strong consistency of θ̃n