On Asymptotic Inference in Stochastic Differential Equations with Time-Varying Covariates

TL;DR

This paper develops asymptotic inference methods for stochastic differential equations with time-varying covariates, covering fixed and random effects, and includes Bayesian approaches, with theoretical validation and empirical demonstrations.

Contribution

It introduces a comprehensive framework for inference in SDEs with time-dependent covariates, including consistency and asymptotic normality results for both ML and Bayesian methods.

Findings

ML estimators are consistent and asymptotically normal.

Bayesian posterior distributions are consistent and asymptotically normal.

Empirical analyses show promising results in real and simulated data.

Abstract

In this article, we introduce a system of stochastic differential equations (SDEs) consisting of time-dependent covariates and consider both fixed and random effects set-ups. We also allow the functional part associated with the drift function to depend upon unknown parameters. In this general set-up of SDE system we establish consistency and asymptotic normality of the M LE through verification of the regularity conditions required by existing relevant theorems. Besides, we consider the Bayesian approach to learning about the population parameters, and prove consistency and asymptotic normality of the corresponding posterior distribution. We supplement our theoretical investigation with simulated and real data analyses, obtaining encouraging results in each case.