Parametric estimation of stochastic differential equations via online gradient descent

TL;DR

This paper introduces an online gradient descent method for estimating parameters in stochastic differential equations from discrete data, providing theoretical risk bounds under model misspecification.

Contribution

It develops a novel online estimation approach with uniform risk bounds for SDEs, incorporating dependent and biased subgradients analysis.

Findings

Provides uniform risk bounds for estimators

Analyzes stochastic mirror descent with dependent subgradients

Ensures ergodicity of diffusion process classes

Abstract

We propose an online parametric estimation method of stochastic differential equations with discrete observations and misspecified modelling based on online gradient descent. Our study provides uniform upper bounds for the risks of the estimators over a family of stochastic differential equations. The derivation of the bounds involves three underlying theoretical results: the analysis of the stochastic mirror descent algorithm based on dependent and biased subgradients, the simultaneous exponential ergodicity of classes of diffusion processes, and the proposal of loss functions whose approximated stochastic subgradients are dependent only on the known model and observations.