Moderate and $L^p$ maximal inequalities for diffusion processes and conformal martingales

TL;DR

This paper establishes sharp moderate maximal inequalities for one-dimensional diffusion processes, extending classical $L^p$ inequalities, and applies these results to various specific and high-dimensional stochastic processes.

Contribution

It introduces the first sharp moderate maximal inequalities for diffusion processes, generalizing classical $L^p$ inequalities and applying them to conformal martingales and high-dimensional processes.

Findings

Established sharp moderate maximal inequalities for diffusion processes.

Extended inequalities to high-dimensional processes like complex OU and conformal martingales.

Applied results to specific processes such as Ornstein-Uhlenbeck and Bessel processes.

Abstract

The $L^p$ maximal inequalities for martingales are one of the classical results in the theory of stochastic processes. Here we establish the sharp moderate maximal inequalities for one-dimensional diffusion processes, which include the $L^p$ maximal inequalities as special cases. Moreover, we apply our theory to many specific examples, including the Ornstein-Uhlenbeck (OU) process, Brownian motion with drift, reflected Brownian motion with drift, Cox-Ingersoll-Ross process, radial OU process, and Bessel process. The results are further applied to establish the moderate maximal inequalities for some high-dimensional processes, including the complex OU process and general conformal local martingales.