Moderate maximal inequalities for the Ornstein-Uhlenbeck process

TL;DR

This paper rigorously proves moderate maximal inequalities for the Ornstein-Uhlenbeck process, extending previous results and providing new inequalities for continuous local martingales, thus enriching the theory of diffusion process inequalities.

Contribution

It provides a rigorous proof of moderate maximum inequalities for the Ornstein-Uhlenbeck process, generalizing previous results and including new inequalities for local martingales.

Findings

Established rigorous proof of moderate maximum inequalities for Ornstein-Uhlenbeck process.

Extended $L^p$ maximum inequalities and included $L^1$ case.

Derived new moderate maximal inequality for continuous local martingales.

Abstract

The maximal inequalities for diffusion processes have drawn increasing attention in recent years. However, the existing proof of the $L^p$ maximum inequalities for the Ornstein-Uhlenbeck process was dubious. Here we give a rigorous proof of the moderate maximum inequalities for the Ornstein-Uhlenbeck process, which include the $L^p$ maximum inequalities as special cases and generalize the remarkable $L^1$ maximum inequalities obtained by Graversen and Peskir [P. Am. Math. Soc., 128(10):3035-3041, 2000]. As a corollary, we also obtain a new moderate maximal inequality for continuous local martingales, which can be viewed as a supplement of the classical Burkholder-Davis-Gundy inequality.