Berry-Esseen bounds for large-time asymptotics of one-dimensional diffusion processes via Malliavin-Stein method

TL;DR

This paper establishes Berry-Esseen bounds for the asymptotic distribution of solutions to certain one-dimensional stochastic differential equations, demonstrating convergence to Gaussian processes with explicit error rates using the Malliavin-Stein method.

Contribution

It provides the first explicit Berry-Esseen bounds for large-time behavior of SDE solutions converging to Gaussian processes, with optimal convergence rates.

Findings

Derived total variation bounds with optimal convergence rates

Applied Malliavin-Stein method for precise error estimation

Showed solutions approximate Gaussian processes as time increases

Abstract

We consider solutions of stochastic differential equations which diverge to infinity as the time parameter goes to infinity. If the coefficients converge as the spacial variable goes to infinity, then the solutions will get close to some Gaussian processes with positive drifts as the time parameter goes to infinity. In this paper, we prove Berry-Esseen type bounds for the solutions in this setting. In particular, we obtain bounds of the total variation distance between the law of the centered and scaled solutions of the stochastic differential equations and the standard normal distribution with an optimal rate of convergence in the time parameter. In the proof we apply the Malliavin-Stein method to estimate the total variation distance.