Discrete-Time Statistical Inference for Multiscale Diffusions

TL;DR

This paper develops and analyzes statistical estimators for multiscale stochastic systems observed through a single slow process, demonstrating their consistency, efficiency, and asymptotic properties both in fixed and high-frequency observation regimes.

Contribution

It introduces new estimators based on a second-order stochastic Taylor expansion for multiscale diffusions and establishes their statistical properties under various observation schemes.

Findings

The minimum contrast estimator (MCE) is consistent, asymptotically normal, and efficient.

The simplified minimum contrast estimator (SMCE) is consistent and asymptotically normal but generally not efficient.

Numerical simulations confirm the theoretical properties of the estimators.

Abstract

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 only. We construct estimators for both averaging and homogenization regimes, based on an appropriate misspecified model motivated by a second-order stochastic Taylor expansion of the slow process with respect to a function of the time-scale separation parameter. In the case of a fixed number of observations, we establish consistency, asymptotic normality, and asymptotic statistical efficiency of a minimum contrast estimator (MCE), the limiting variance having been identified explicitly; we furthermore establish consistency and asymptotic normality of a simplified minimum constrast estimator (SMCE), which is however not in general efficient. These results are then extended to the case of high-frequency observations under a condition restricting the rate at which the number of observations may grow vis-\`a-vis the separation of scales. Numerical simulations illustrate the theoretical results.