Drift Estimation for Stochastic Differential Equations with Denoising Diffusion Models

TL;DR

This paper introduces a novel drift estimation method for multivariate stochastic differential equations using denoising diffusion models, demonstrating competitive performance across various dimensions.

Contribution

It formulates drift estimation as a denoising problem and develops a conditional diffusion model that can simulate trajectories, offering a new approach to SDE parameter estimation.

Findings

Matches classical methods in low dimensions

Remains competitive in higher dimensions

Gains are not solely due to architecture design

Abstract

We study the estimation of time-homogeneous drift functions in multivariate stochastic differential equations with known diffusion coefficient, from multiple trajectories observed at high frequency over a fixed time horizon. We formulate drift estimation as a denoising problem conditional on previous observations, and propose an estimator of the drift function which is a by-product of training a conditional diffusion model capable of simulating new trajectories dynamically. Across different drift classes, the proposed estimator was found to match classical methods in low dimensions and remained consistently competitive in higher dimensions, with gains that cannot be attributed to architectural design choices alone.