🤖 AI Summary
This study addresses the estimation of the time-homogeneous drift function in multivariate stochastic differential equations (SDEs) with known diffusion coefficients, based on high-frequency observations from multiple trajectories. To this end, the authors propose formulating drift estimation as a conditional denoising problem conditioned on historical observations and introduce a conditional diffusion model that dynamically generates new trajectories from which the drift estimator is extracted. This approach represents the first application of conditional denoising diffusion models to drift estimation in SDEs. It significantly outperforms classical methods in high-dimensional settings without relying on any specific neural network architecture, while achieving comparable performance to existing approaches in low-dimensional cases, thereby demonstrating both its effectiveness and scalability.
📝 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.