Semi-Parametric Drift and Diffusion Estimation for Multiscale Diffusions

TL;DR

This paper introduces a semi-parametric estimation method for identifying parameters in the effective dynamics of multiscale diffusions, overcoming limitations of classical estimators, and demonstrates its effectiveness through extensive simulations.

Contribution

The paper presents a novel semi-parametric algorithm for estimating drift and diffusion parameters in multiscale diffusions, addressing shortcomings of traditional methods.

Findings

The proposed algorithm accurately estimates parameters in multiscale diffusion models.

Classical estimators like MLE and quadratic variation fail in multiscale settings.

Numerical simulations show the method provides unbiased, reliable estimates.

Abstract

We consider the problem of statistical inference for the effective dynamics of multiscale diffusion processes with (at least) two widely separated characteristic time scales. More precisely, we seek to determine parameters in the effective equation describing the dynamics on the longer diffusive time scale, i.e. in a homogenization framework. We examine the case where both the drift and the diffusion coefficients in the effective dynamics are space-dependent and depend on multiple unknown parameters. It is known that classical estimators, such as Maximum Likelihood and Quadratic Variation of the Path Estimators, fail to obtain reasonable estimates for parameters in the effective dynamics when based on observations of the underlying multiscale diffusion. We propose a novel algorithm for estimating both the drift and diffusion coefficients in the effective dynamics based on a semi-parametric framework. We demonstrate by means of extensive numerical simulations of a number of selected examples that the algorithm performs well when applied to data from a multiscale diffusion. These examples also illustrate that the algorithm can be used effectively to obtain accurate and unbiased estimates.