sde-learn

Overview

This repository provides a Python implementation of the method proposed by Bonalli et al., 2023, for estimating the coefficients of multidimensional nonlinear stochastic differential equations (SDEs). This method divides the estimation into two steps:

1. Density estimation. Estimation of the probability density associated with the unknown SDE and its derivatives using a dataset of sample paths.
2. Coefficients estimation with Fokker-Planck matching. Estimation of the SDE coefficients (drift and diffusion) by matching the Fokker-Planck dynamics described by the estimated density.

For a detailed description of the method, please refer to Bonalli et al., 2023.

Features. This code implements the following features.

• Kernel-based modeling. Both steps employ kernel-based models for estimating the probability density and the SDE coefficients.
• Positivity constraints. To ensure physical and mathematical validity, it is crucial to enforce the positive-definiteness of the estimated diffusion coefficient. This is achieved by considering uniform diffusion and applying positivity constraints at a finite set of points.
• Fast computations. The code implements Nyström approximation to reduce computational complexity with large datasets. Additionally, for the Gaussian kernel, it enables fast computation of all partial derivatives and Gram matrices through vectorized computations in Python using NumPy.

Examples

The examples folder contains several scripts demonstrating different applications of the sde-learn library.

1. example_ornstein_uhlenbeck_paths_plot.py. Illustrates the generation and plotting of sample paths from an Ornstein-Uhlenbeck process.
2. example_kde_plot.py. Demonstrates the use of the ProbaDensityEstimator for estimating and visualizing the probability density of sample paths from an SDE.
3. example_sde_identification_1d.py. Offers a complete example for simulating a one-dimensional SDE, estimating its density, and subsequently estimating its coefficients using Fokker-Planck matching.
4. example_sde_identification_2d_1.py. Provides a complete example for a two-dimensional nonlinear SDE, covering the estimation of its density and coefficients.
5. example_sde_identification_2d_2.py. Similar to the previous example, focusing on a different two-dimensional nonlinear SDE.

Installation

To install:

1. Clone the repository.

   git clone https://github.com/lmotte/sde-learn.git

2. Install the required dependencies (Python 3.x required).

   pip install -r requirements.txt

License

This project is licensed under the MIT License - see the LICENSE file for details.

References

• Bonalli, Riccardo, and Alessandro Rudi. "Non-Parametric Learning of Stochastic Differential Equations with Fast Rates of Convergence." arXiv preprint arXiv:2305.15557 (2023).