Adaptive Sub-sampling for Parametric Estimation of Gaussian Diffusions

Abstract

We consider a Gaussian diffusion X _t (Ornstein-Uhlenbeck process) with drift coefficient γ and diffusion coefficient σ ², and an approximating process $Y^{\varepsilon}_{t}$ converging to X _t in L ₂ as ε→0. We study estimators $\hat{\gamma}_{\varepsilon}$, $\hat{\sigma}^{2}_{\varepsilon}$ which are asymptotically equivalent to the Maximum likelihood estimators of γ and σ ², respectively. We assume that the estimators are based on the available N=N(ε) observations extracted by sub-sampling only from the approximating process $Y^{\varepsilon}_{t}$ with time step Δ=Δ(ε). We characterize all such adaptive sub-sampling schemes for which $\hat{\gamma}_{\varepsilon}$, $\hat{\sigma}^{2}_{\varepsilon}$ are consistent and asymptotically efficient estimators of γ and σ ² as ε→0. The favorable adaptive sub-sampling schemes are identified by the conditions ε→0, Δ→0, (Δ/ε)→∞, and NΔ→∞, which implies that we sample from the process $Y^{\varepsilon}_{t}$ with a vanishing but coarse time step Δ(ε)≫ε. This study highlights the necessity to sub-sample at adequate rates when the observations are not generated by the underlying stochastic model whose parameters are being estimated. The adequate sub-sampling rates we identify seem to retain their validity in much wider contexts such as the additive triad application we briefly outline.