Parametric inference for discretely observed multidimensional diffusions with small diffusion coefficient

TL;DR

This paper develops and compares parametric inference methods for multidimensional diffusions observed discretely, focusing on small diffusion coefficients, and demonstrates their effectiveness through simulations and epidemiological applications.

Contribution

It introduces consistent and asymptotically normal estimators for drift and diffusion parameters in discretely observed diffusions with small noise, extending existing methods.

Findings

Estimators are consistent and asymptotically normal as noise and observation intervals decrease.

Simulation studies compare various estimation methods under different parameter regimes.

Application to epidemiology shows practical relevance of the proposed methods.

Abstract

We consider a multidimensional diffusion X with drift coefficient b({\alpha},X(t)) and diffusion coefficient {\epsilon}{\sigma}({\beta},X(t)). The diffusion is discretely observed at times t_k=k{\Delta} for k=1..n on a fixed interval [0,T]. We study minimum contrast estimators derived from the Gaussian process approximating X for small {\epsilon}. We obtain consistent and asymptotically normal estimators of {\alpha} for fixed {\Delta} and {\epsilon}\rightarrow0 and of ({\alpha},{\beta}) for {\Delta}\rightarrow0 and {\epsilon}\rightarrow0. We compare the estimators obtained with various methods and for various magnitudes of {\Delta} and {\epsilon} based on simulation studies. Finally, we investigate the interest of using such methods in an epidemiological framework.