Non-parametric estimation of Stochastic Differential Equations from stationary time-series

TL;DR

This paper develops and analyzes non-parametric estimators for drift and diffusion coefficients of stochastic differential equations from stationary time-series data, optimizing convergence and computational efficiency.

Contribution

It introduces new estimators based on conditional expectations and provides a detailed analysis of their consistency, error bounds, and optimal parameter relationships.

Findings

Estimators are consistent and have quantifiable mean squared error.

Optimal relationships between data points and discretization parameters are derived.

Numerical simulations confirm theoretical results.

Abstract

We study efficiency of non-parametric estimation of diffusions (stochastic differential equations driven by Brownian motion) from long stationary trajectories. First, we introduce estimators based on conditional expectation which is motivated by the definition of drift and diffusion coefficients. These estimators involve time- and space-discretization parameters for computing expected values from discretely-sampled stationary data. Next, we analyze consistency and mean squared error of these estimators depending on computational parameters. We derive relationships between the number of observational points, time- and space-discretization parameters in order to achieve the optimal speed of convergence and minimize computational complexity. We illustrate our approach with numerical simulations.