Stationary Density Estimation of Itô Diffusions Using Deep Learning

Abstract

In this paper, we consider the density estimation problem associated with the stationary measure of ergodic Itô diffusions from a discrete-time series that approximate the solutions of the stochastic differential equations. To take 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–Planck 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 PDE regression solution obtained with neural-network models. We demonstrate the effectiveness of this method by numerical simulations of a two-dimensional Student t-distribution and a 20-dimensional Langevin dynamics.