Perturbation-based inference for diffusion processes: Obtaining effective models from multiscale data

TL;DR

This paper develops a new convergent inference method for stochastic differential equations based on perturbation techniques, enabling effective parameter estimation from multiscale data where traditional methods fail.

Contribution

It introduces a novel inference procedure that converges for multiscale stochastic models, addressing stability issues of standard techniques like maximum likelihood.

Findings

The new method is stable and convergent for multiscale stochastic models.

Standard maximum likelihood estimators are unstable in this setting.

Numerical examples demonstrate the effectiveness of the proposed approach.

Abstract

We consider the inference problem for parameters in stochastic differential equation models from discrete time observations (e.g. experimental or simulation data). Specifically, we study the case where one does not have access to observations of the model itself, but only to a perturbed version which converges weakly to the solution of the model. Motivated by this perturbation argument, we study the convergence of estimation procedures from a numerical analysis point of view. More precisely, we introduce appropriate consistency, stability, and convergence concepts and study their connection. It turns out that standard statistical techniques, such as the maximum likelihood estimator, are not convergent methodologies in this setting, since they fail to be stable. Due to this shortcoming, we introduce and analyse a novel inference procedure for parameters in stochastic differential equation models which turns out to be convergent. As such, the method is particularly suited for the estimation of parameters in effective (i.e. coarse-grained) models from observations of the corresponding multiscale process. We illustrate these theoretical findings via several numerical examples.