Bounding the Solutions to Some SDEs via Ergodic Theory

TL;DR

This paper develops a method to find optimal bounds for the supremum norm of solutions to autonomous SDEs with smooth invariant measures, using ergodic theory, applicable in many cases including one-dimensional SDEs.

Contribution

It introduces a novel approach leveraging ergodic theory to establish bounds on SDE solutions, often optimal, with special considerations for measure-zero sets in one-dimensional cases.

Findings

Provides bounds for the supremum of SDE solutions over time intervals.

Bounds are often optimal and hold almost everywhere.

In one-dimensional cases, measure-zero sets can be uniform for all initial conditions.

Abstract

In this note we consider autonomous SDEs admitting smooth invariant measures. We present a method in finding (almost everywhere) good bounds for $\sup \{\|X_t\|: t \in [0, T]\}$ for strong solutions $X_{\cdot}$ to such SDEs, which in many cases are optimal bounds. In some situation (especially in one-dimensional SDEs' cases), the discarded measure-zero set can be chosen to be a measure-zero set of the underlying Brownian motion uniform for all initial points $X_0=x$.