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

Xian Chen; Yong Chen; Mumien Cheng; Chen Jia

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arXiv · 2111.02641

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

Xian Chen·Yong Chen·Mumien Cheng·Chen Jia

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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.

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Xian Chen, Yong Chen, Mumien Cheng, Chen Jia. 2021-11-04. Moderate and $L^p$ maximal inequalities for diffusion processes and conformal martingales. https://arxiv.org/abs/2111.02641

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