Stationary Density Estimation of It\^o Diffusions Using Deep Learning

TL;DR

This paper introduces a deep learning approach to estimate the stationary density of ergodic Itô diffusions by combining neural network regression of SDE coefficients with PDE-based density estimation, supported by theoretical convergence analysis.

Contribution

It presents a novel neural network framework for density estimation of Itô diffusions that integrates drift and diffusion estimation with PDE solving, with proven convergence guarantees.

Findings

Successfully estimated densities for 2D Student's t distribution.

Extended method to high-dimensional Langevin dynamics (20D).

Demonstrated theoretical convergence and error bounds.

Abstract

In this paper, we consider the density estimation problem associated with the stationary measure of ergodic It\^o diffusions from a discrete-time series that approximate the solutions of the stochastic differential equations. To take an advantage of the characterization of density function through the stationary solution of a parabolic-type Fokker-Planck PDE, we proceed as follows. First, we employ deep neural networks to approximate the drift and diffusion terms of the SDE by solving appropriate supervised learning tasks. Subsequently, we solve a steady-state Fokker-Plank equation associated with the estimated drift and diffusion coefficients with a neural-network-based least-squares method. We establish the convergence of the proposed scheme under appropriate mathematical assumptions, accounting for the generalization errors induced by regressing the drift and diffusion coefficients, and the PDE solvers. This theoretical study relies on a recent perturbation theory of Markov chain result that shows a linear dependence of the density estimation to the error in estimating the drift term, and generalization error results of nonparametric regression and of PDE regression solution obtained with neural-network models. The effectiveness of this method is reflected by numerical simulations of a two-dimensional Student's t distribution and a 20-dimensional Langevin dynamics.