A review on asymptotic inference in stochastic differential equations with mixed-effects

TL;DR

This survey reviews recent theoretical advances in asymptotic inference for stochastic differential equations with mixed-effects, focusing on estimators' properties in specific models with high-frequency data.

Contribution

It provides a comprehensive overview of the asymptotic properties of estimators in mixed-effects SDE models, highlighting recent theoretical developments.

Findings

Asymptotic properties of estimators are established for specific classes of mixed-effects SDEs.

Explicit estimators are analyzed under high-frequency observation schemes.

Theoretical inference is primarily developed for models observed without measurement error.

Abstract

This paper is a survey of recent contributions on estimation in stochastic differential equations with mixed-effects. These models involve N stochastic differential equations with common drift and diffusion functions but random parameters that allow for differences between processes. The main objective is to estimate the distribution of the random effects and possibly other fixed parameters that are common to the N processes. While many algorithms have been proposed, the theoretical aspects related to estimation have been little studied. This review article focuses only on theoretical inference for stochastic differential equations with mixed-effects. It has so far only been considered in some very specific classes of mixed-effect diffusion models, observed without measurement error, where explicit estimators can be defined. Within this framework, the asymptotic properties of several estimators, either parametric or nonparametric, are discussed. Different schemes of observations are considered according to the approach, associating a large number of individuals with, in most cases, high-frequency observations of the trajectories.