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

Kavli Affiliate: Xian Chen

| First 5 Authors: Xian Chen, Yong Chen, Mumien Cheng, Chen Jia,

| Summary:

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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