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Moderate and $$L^p$$ L p Maximal Inequalities for Diffusion Processes and Conformal Martingales

Author

Listed:
  • Xian Chen

    (Xiamen University)

  • Yong Chen

    (Jiangxi Normal University)

  • Yumin Cheng

    (Jiangxi Normal University)

  • Chen Jia

    (Beijing Computational Science Research Center)

Abstract

The $$L^p$$ 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 generalize the $$L^p$$ L p maximal inequalities for diffusions. 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.

Suggested Citation

  • Xian Chen & Yong Chen & Yumin Cheng & Chen Jia, 2024. "Moderate and $$L^p$$ L p Maximal Inequalities for Diffusion Processes and Conformal Martingales," Journal of Theoretical Probability, Springer, vol. 37(4), pages 2990-3014, November.
    • Handle: RePEc:spr:jotpro:v:37:y:2024:i:4:d:10.1007_s10959-024-01359-9
    DOI: 10.1007/s10959-024-01359-9
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    References listed on IDEAS

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    1. Chen Jia, 2019. "Sharp Moderate Maximal Inequalities for Upward Skip-Free Markov Chains," Journal of Theoretical Probability, Springer, vol. 32(3), pages 1382-1398, September.
    2. Chen, Xian & Jia, Chen, 2020. "Mathematical foundation of nonequilibrium fluctuation–dissipation theorems for inhomogeneous diffusion processes with unbounded coefficients," Stochastic Processes and their Applications, Elsevier, vol. 130(1), pages 171-202.
    3. Goran Peskir, 2001. "Bounding the Maximal Height of a Diffusion by the Time Elapsed," Journal of Theoretical Probability, Springer, vol. 14(3), pages 845-855, July.
    4. S. E. Graversen & G. Peškir, 1998. "Optimal Stopping and Maximal Inequalities for Linear Diffusions," Journal of Theoretical Probability, Springer, vol. 11(1), pages 259-277, January.
    5. Jia, Chen, 2016. "A solution to the reversible embedding problem for finite Markov chains," Statistics & Probability Letters, Elsevier, vol. 116(C), pages 122-130.
    6. Zhang, Xicheng, 2005. "Strong solutions of SDES with singular drift and Sobolev diffusion coefficients," Stochastic Processes and their Applications, Elsevier, vol. 115(11), pages 1805-1818, November.
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