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

Abstract

We study the drift parameter estimation problem for a fractional Vasicek-typemodel X:={X_t,t⩾0}, that is defined as dX_t=θ(µ+X_t)dt+dB_t^H, t⩾0 withunknown parameters θ>0 and µ∈ℝ, where {B_t^H,t⩾0}is a fractional Brownianmotion of Hurst index H ∈]0, 1[. Let (θ_t ) ̂and (µ_t ) ̂be the least squares-type estimatorsof θand μ, respectively, based on continuous observation of X. In this paper weassume that the process {X_t,t⩾0}is observed at discrete time instants t_i=iΔ_n,i=1,...,n. We analyze discrete versions (θ_n ) ̃and (µ_n ) ̃for (θ_t ) ̂and (µ_t ) ̂ respectively. We show that the sequence √(〖nΔ〗_n ) ((θ_n ) ̃-θ) is tight and √(〖nΔ〗_n )((μ_n ) ̃-μ) is not tight. Moreover, we prove the strong consistency of (θ_n ) ̃ .