Parameter estimation for Gaussian mean-reverting Ornstein–Uhlenbeck processes of the second kind: Non-ergodic case
Abstract
We consider a least square-type method to estimate the drift parameters for the mean-reverting Ornstein–Uhlenbeck process of the second kind {Xt,t≥0} defined as dXt=𝜃(μ+Xt)dt+dY(1)t,G,t≥0, with unknown parameters 𝜃>0 and μ∈ℝ, where Y(1)t,G:=∫t0e−sdGas with at=HetH, and {Gt,t≥0} is a Gaussian process. In order to establish the consistency and the asymptotic distribution of least square-type estimators of 𝜃 and μ based on the continuous-time observations {Xt,t∈[0,T]} as T→∞, we impose some technical conditions on the process G, which are satisfied, for instance, if G is a fractional Brownian motion with Hurst parameter H∈(0,1), G is a subfractional Brownian motion with Hurst parameter H∈(0,1) or G is a bifractional Brownian motion with Hurst parameters (H,K)∈(0,1)×(0,1]. Our method is based on pathwise properties of {Xt,t≥0} and {Y(1)t,G,t≥0} proved in the sequel.
