# Non-parametric Estimation of Stochastic Differential Equations with Sparse Gaussian Processes

## TL;DR

This paper presents a non-parametric approach using sparse Gaussian processes to estimate drift and diffusion functions of SDEs from densely observed data, enabling efficient modeling of complex temporal dynamics.

## Contribution

It introduces a novel sparse Gaussian process-based method for non-parametric SDE estimation, improving computational efficiency and applicability to real-world data.

## Key findings

- Accurately estimates SDE components from simulated data
- Effectively captures complex dynamics in economic and paleoclimatic data
- Demonstrates computational efficiency with sparse approximation

## Abstract

The application of Stochastic Differential Equations (SDEs) to the analysis of temporal data has attracted increasing attention, due to their ability to describe complex dynamics with physically interpretable equations. In this paper, we introduce a non-parametric method for estimating the drift and diffusion terms of SDEs from a densely observed discrete time series. The use of Gaussian processes as priors permits working directly in a function-space view and thus the inference takes place directly in this space. To cope with the computational complexity that requires the use of Gaussian processes, a sparse Gaussian process approximation is provided. This approximation permits the efficient computation of predictions for the drift and diffusion terms by using a distribution over a small subset of pseudo-samples. The proposed method has been validated using both simulated data and real data from economy and paleoclimatology. The application of the method to real data demonstrates its ability to capture the behaviour of complex systems.