Moderate maximal inequalities for the Ornstein-Uhlenbeck process
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- by Chen Jia and Guohuan Zhao PDF
- Proc. Amer. Math. Soc. 148 (2020), 3607-3615 Request permission
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.References
- G. E. Uhlenbeck and L. S. Ornstein, On the theory of the Brownian motion, Phys. Rev. 36 (1930), 823.
- Edward Nelson, Dynamical theories of Brownian motion, Princeton University Press, Princeton, N.J., 1967. MR 0214150
- Xian Chen and Chen Jia, Mathematical foundation of nonequilibrium fluctuation-dissipation theorems for inhomogeneous diffusion processes with unbounded coefficients, Stochastic Process. Appl. 130 (2020), no. 1, 171–202. MR 4035027, DOI 10.1016/j.spa.2019.02.005
- Daniel Revuz and Marc Yor, Continuous martingales and Brownian motion, 3rd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 293, Springer-Verlag, Berlin, 1999. MR 1725357, DOI 10.1007/978-3-662-06400-9
- S. E. Graversen and G. Peškir, Maximal inequalities for Bessel processes, J. Inequal. Appl. 2 (1998), no. 2, 99–119. MR 1671984, DOI 10.1155/S102558349800006X
- S. E. Graversen and G. Peskir, Optimal stopping and maximal inequalities for geometric Brownian motion, J. Appl. Probab. 35 (1998), no. 4, 856–872. MR 1671236, DOI 10.1239/jap/1032438381
- S. E. Graversen and G. Peskir, Maximal inequalities for the Ornstein-Uhlenbeck process, Proc. Amer. Math. Soc. 128 (2000), no. 10, 3035–3041. MR 1664394, DOI 10.1090/S0002-9939-00-05345-4
- Goran Peskir, Bounding the maximal height of a diffusion by the time elapsed, J. Theoret. Probab. 14 (2001), no. 3, 845–855. MR 1860525, DOI 10.1023/A:1017505509361
- Litan Yan, Ligang Lu, and Zhiqiang Xu, $L^p$ estimates on a time-inhomogeneous diffusion process, J. Math. Phys. 46 (2005), no. 8, 083513, 8. MR 2165859, DOI 10.1063/1.2000208
- Litan Yan and Bei Zhu, A ratio inequality for Bessel processes, Statist. Probab. Lett. 66 (2004), no. 1, 35–44. MR 2025651, DOI 10.1016/j.spl.2003.10.003
- Litan Yan and Bei Zhu, $L^p$-estimates on diffusion processes, J. Math. Anal. Appl. 303 (2005), no. 2, 418–435. MR 2122226, DOI 10.1016/j.jmaa.2004.08.029
- Ya. A. Lyulko and A. N. Shiryaev, Sharp maximal inequalities for stochastic processes, Proc. Steklov Inst. Math. 287 (2014), no. 1, 155–173. Translation of Tr. Mat. Inst. Steklova 287 (2014), 162–181. MR 3484328, DOI 10.1134/S0081543814080100
- Xian Chen and Chen Jia, Identification of unstable fixed points for randomly perturbed dynamical systems with multistability, J. Math. Anal. Appl. 446 (2017), no. 1, 521–545. MR 3554742, DOI 10.1016/j.jmaa.2016.07.060
- Chen Jia, Sharp moderate maximal inequalities for upward skip-free Markov chains, J. Theoret. Probab. 32 (2019), no. 3, 1382–1398. MR 3979672, DOI 10.1007/s10959-018-0820-6
- D. L. Burkholder, Distribution function inequalities for martingales, Ann. Probability 1 (1973), 19–42. MR 365692, DOI 10.1214/aop/1176997023
Additional Information
- Chen Jia
- Affiliation: Applied and Computational Mathematics Division, Beijing Computational Science Research Center, Beijing 100193, People’s Republic of China
- MR Author ID: 976605
- Email: chenjia@csrc.ac.cn
- Guohuan Zhao
- Affiliation: Faculty of Mathematics, Bielefeld University, Bielefeld 33615, Germany
- MR Author ID: 1084395
- ORCID: 0000-0003-4523-6239
- Email: zhaoguohuan@gmail.com
- 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
- © Copyright 2020 American Mathematical Society
- 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
- MathSciNet review: 4108864