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On the exponential integrability of the derivative of intersection and self-intersection local time for Brownian motion and related processes

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  • Das, Kaustav
  • Markowsky, Gregory
  • Wu, Binghao

Abstract

We show that the derivative of the intersection and self-intersection local times of alpha-stable processes are exponentially integrable for certain parameter values. This includes the Brownian motion case. We also discuss related results present in the literature for fractional Brownian motion, and in particular give a counter-example to a result in Guo et al. (2019) related to this question.

Suggested Citation

  • Das, Kaustav & Markowsky, Gregory & Wu, Binghao, 2025. "On the exponential integrability of the derivative of intersection and self-intersection local time for Brownian motion and related processes," Stochastic Processes and their Applications, Elsevier, vol. 183(C).
    • Handle: RePEc:eee:spapps:v:183:y:2025:i:c:s030441492500033x
    DOI: 10.1016/j.spa.2025.104592
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    References listed on IDEAS

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    1. Jingjun Guo & Yaozhong Hu & Yanping Xiao, 2019. "Higher-Order Derivative of Intersection Local Time for Two Independent Fractional Brownian Motions," Journal of Theoretical Probability, Springer, vol. 32(3), pages 1190-1201, September.
    2. Qian Yu & Qiangqiang Chang & Guangjun Shen, 2023. "Smoothness of higher order derivative of self-intersection local time for fractional Brownian motion," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 52(10), pages 3541-3556, May.
    3. Qian Yu, 2021. "Higher-Order Derivative of Self-Intersection Local Time for Fractional Brownian Motion," Journal of Theoretical Probability, Springer, vol. 34(4), pages 1749-1774, December.
    4. Hong, Minhao & Xu, Fangjun, 2021. "Derivatives of local times for some Gaussian fields II," Statistics & Probability Letters, Elsevier, vol. 172(C).
    5. Jingjun Guo & Cuiyun Zhang & Aiqin Ma, 2024. "Derivative of Multiple Self-intersection Local Time for Fractional Brownian Motion," Journal of Theoretical Probability, Springer, vol. 37(1), pages 623-641, March.
    6. Yan, Litan & Yu, Xianye & Chen, Ruqing, 2017. "Derivative of intersection local time of independent symmetric stable motions," Statistics & Probability Letters, Elsevier, vol. 121(C), pages 18-28.
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