Parametric Estimation of Stationary Stochastic Processes Under Indirect Observability

Abstract

For many natural turbulent dynamic systems, observed high dimensional dynamic data can be approximated at slow time scales by a process X _t driven by a systems of stochastic differential equations (SDEs). When one tries to estimate the parameters of this unobservable SDEs systems, there is a clear mismatch between the available data and the SDEs dynamics to be parametrized. Here, we formalize this Indirect Observability framework as follows.

We consider an unobservable centered stationary Gaussian process X _t with covariance function K(u,θ)=E[X _t X _t+u ], parametrized by an unknown vector θ which lies in a compact subset Θ of ℝ^p. We assume that the only observable data are generated by centered stationary processes $Y_{t}^{\varepsilon }$, indexed by a scale separation parameter ε>0. These approximating processes have arbitrary probability distributions, exponentially decaying covariances, and are assumed to converge to X _t in L ₄ as ε→0. We show how to construct estimators of the underlying parameter vector θ which depend only on the observable data $Y_{t}^{\varepsilon }$, and converge to the true parameter values as ε→0.

We study adaptive subsampling schemes involving [N(ε)+k(ε)]→∞ observations $V_{n} = Y^{\varepsilon }_{n \Delta(\varepsilon )}$ extracted from the approximating process $Y^{\varepsilon }_{t}$ by subsampling at time intervals Δ(ε)→0. We focus on parameter estimators which are smooth functions of subsampled empirical covariance estimators $\hat{r}_{k}(N,\Delta)$ associated to non vanishing time lags k(ε)Δ(ε) tending to fixed positive limits as ε→0.

We show that provided lim  _ε→0 N(ε)Δ(ε)=+∞, these subsampled approximate covariance estimators converge in L ₂ to the true covariance function K(u,θ) of X _t for all u,θ. Applying a generic version of the method of moments suitably boosted up by adequately adjusted multiple subsampling schemes, we show that this implies, in a very wide range of situations, the existence of consistent estimators $\hat{\theta}(\varepsilon )$ of the unknown parameter vector θ, based only on adequately subsampled approximate data $Y^{\varepsilon }_{t}$.