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Moderate maximal inequalities for the Ornstein-Uhlenbeck process


Authors: Chen Jia and Guohuan Zhao
Journal: Proc. Amer. Math. Soc. 148 (2020), 3607-3615
MSC (2010): Primary 60H10, 60J60, 60J65, 60G44, 60E15
DOI: https://doi.org/10.1090/proc/14804
Published electronically: April 28, 2020
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Abstract: The maximal inequalities for diffusion processes have drawn increasing attention in recent years. Here we prove the moderate maximal inequality for the Ornstein-Uhlenbeck process, which includes the $ L^p$ maximal inequality as a special case and generalizes the $ L^1$ maximal inequality obtained by Graversen and Peskir [Proc. Amer. 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 an extension of the classical Burkholder-Davis-Gundy inequality.


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Additional Information

Chen Jia
Affiliation: Applied and Computational Mathematics Division, Beijing Computational Science Research Center, Beijing 100193, People’s Republic of China
Email: chenjia@csrc.ac.cn

Guohuan Zhao
Affiliation: Faculty of Mathematics, Bielefeld University, Bielefeld 33615, Germany
Email: zhaoguohuan@gmail.com

DOI: https://doi.org/10.1090/proc/14804
Keywords: Moderate function, law of the iterated logarithm, good $\lambda$ inequality, Brownian motion, local martingale, Burkholder-Davis-Gundy inequality
Received by editor(s): November 17, 2017
Received by editor(s) in revised form: May 28, 2018
Published electronically: April 28, 2020
Additional Notes: The first author acknowledges support from the NSAF grant from the National Natural Science Foundation of China (NSFC) with grant No. U1930402
The second author acknowledges support from the National Postdoctoral Program for Innovative Talents of China (No. 201600182)
The second author is the corresponding author
Communicated by: Zhen-Qing Chen
Article copyright: © Copyright 2020 American Mathematical Society