Discrete-Time Statistical Inference for Multiscale Diffusions in the Averaging and Homogenization Regime

We study statistical inference for small-noise-perturbed multiscale dynamical systems under the assumption that we observe a single time series from the slow process. We study both averaging and homogenization regimes, constructing statistical estimators which we prove to be consistent, asymptotically normal (with explicit characterization of the limiting variance), and, in certain cases, asymptotically efficient. In the case of a fixed number of observations the proposed methods produce consistent and asymptotically normal estimates, making the results readily applicable. For high-frequency observations, we prove consistency and asymptotic normality under a condition restricting the rate at which the number of observations may grow vis-\`a-vis the separation of scales. The estimators are based on an appropriate misspecified model motivated by a second-order stochastic Taylor expansion of the slow component with respect to a function of the time-scale separation parameter. Numerical simulations illustrate the theoretical results.