Nonparametric estimation of stochastic differential equations with sparse Gaussian processes.

Area of Science:

• Statistics
• Time Series Analysis
• Computational Mathematics

Background:

• Stochastic Differential Equations (SDEs) are increasingly used for analyzing temporal data due to their ability to model complex dynamics.
• Estimating the drift and diffusion terms of SDEs from discrete time series is challenging.
• Gaussian processes offer a powerful framework for nonparametric inference in function spaces.

Purpose of the Study:

• To introduce a novel nonparametric method for estimating drift and diffusion terms of SDEs.
• To enable direct inference in function space using Gaussian processes.
• To provide an efficient approximation for computational tractability.

Main Methods:

• Utilizing Gaussian processes as priors for nonparametric estimation.
• Implementing a sparse Gaussian process approximation to handle computational complexity.
• Applying the method to densely observed discrete time series data.

Main Results:

• The proposed method accurately estimates drift and diffusion terms of SDEs.
• Sparse Gaussian process approximation enables efficient prediction using pseudosamples.
• Validation on simulated and real-world data (economy, paleoclimatology) demonstrates effectiveness.

Conclusions:

• The developed nonparametric method provides an effective tool for analyzing complex temporal dynamics described by SDEs.
• The sparse Gaussian process approximation enhances computational efficiency for SDE analysis.
• The method shows promise for applications in diverse scientific fields requiring time series modeling.