Parametric Estimation of Stationary Stochastic Processes under Indirect Observability

For many natural turbulent dynamic systems, observed high dimensional dynamic data can be approximated at slow time scales by a process Xt 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 Xt with covariance function K(u, θ) = E[XtXt+u], parametrized by an unknown vector θ which lies in a compact subset Θ of R. We assume that the only observable data are generated by centered stationary processes Y ε t , indexed by a scale separation parameter ε > 0. These approximating processes have arbitrary probability distributions, exponentially decaying covariances, and are assumed to converge to Xt in L4 as ε → 0. We show how to construct R. Azencott University of Houston Department of Mathematics Houston, TX 77204-3008 Emeritus Professor Ecole Normale Superieure, Paris, France E-mail: [email protected] A. Beri University of Houston Department of Mathematics Houston, TX 77204-3008 E-mail: [email protected] I. Timofeyev University of Houston Department of Mathematics Houston, TX 77204-3008 E-mail: [email protected]