IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v176y2024ics0304414924001388.html

Sample path moderate deviations for shot noise processes in the high intensity regime

Author

Listed:
  • Anugu, Sumith Reddy
  • Pang, Guodong

Abstract

We study the sample-path moderate deviation principle (MDP) for shot noise processes in the high intensity regime. The shot noise processes have a renewal arrival process, non-stationary noises (with arrival-time dependent distributions) and a general shot response function of the noises. The rate function in the MDP exhibits a memory phenomenon in this asymptotic regime, which is in contrast with that in the conventional time–space scaling regime. To prove the sample-path MDP, we first establish that this is equivalent to establishing the sample-path MDP of another process that is easier to study. We prove its finite-dimensional MDP and then establish the exponential tightness under the Skorohod J1 topology. This results in the sample-path MDP in D under the Skorohod J1 topology with a rate function that is derived from the rate function in the finite-dimensional MDP using the tools of reproducing kernel Hilbert space. In the proofs, because of the non-stationarity of shot noise process, we establish a new exponential maximal inequality and use it to prove exponential tightness and the aforementioned equivalence.

Suggested Citation

  • Anugu, Sumith Reddy & Pang, Guodong, 2024. "Sample path moderate deviations for shot noise processes in the high intensity regime," Stochastic Processes and their Applications, Elsevier, vol. 176(C).
    • Handle: RePEc:eee:spapps:v:176:y:2024:i:c:s0304414924001388
    DOI: 10.1016/j.spa.2024.104432
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304414924001388
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spa.2024.104432?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to

    for a different version of it.

    References listed on IDEAS

    as
    1. Claudio Macci & Gabriele Stabile & Giovanni Luca Torrisi, 2005. "Lundberg parameters for non standard risk processes," Scandinavian Actuarial Journal, Taylor & Francis Journals, vol. 2005(6), pages 417-432.
    2. Solesne Bourguin & Thanh Dang & Konstantinos Spiliopoulos, 2023. "Moderate Deviation Principle for Multiscale Systems Driven by Fractional Brownian Motion," Journal of Theoretical Probability, Springer, vol. 36(1), pages 1-57, March.
    3. Pang, Guodong & Zhou, Yuhang, 2018. "Functional limit theorems for a new class of non-stationary shot noise processes," Stochastic Processes and their Applications, Elsevier, vol. 128(2), pages 505-544.
    4. Jacquier, Antoine & Pannier, Alexandre, 2022. "Large and moderate deviations for stochastic Volterra systems," Stochastic Processes and their Applications, Elsevier, vol. 149(C), pages 142-187.
    5. Antoine Jacquier & Alexandre Pannier, 2020. "Large and moderate deviations for stochastic Volterra systems," Papers 2004.10571, arXiv.org, revised Apr 2022.
    6. Iglehart, Donald L., 1973. "Weak convergence of compound stochastic process, I," Stochastic Processes and their Applications, Elsevier, vol. 1(1), pages 11-31, January.
    7. Onno Boxma & Michel Mandjes, 2021. "Shot-noise queueing models," Queueing Systems: Theory and Applications, Springer, vol. 99(1), pages 121-159, October.
    8. Iksanov, Alexander, 2013. "Functional limit theorems for renewal shot noise processes with increasing response functions," Stochastic Processes and their Applications, Elsevier, vol. 123(6), pages 1987-2010.
    9. Budhiraja, Amarjit & Chen, Jiang & Dupuis, Paul, 2013. "Large deviations for stochastic partial differential equations driven by a Poisson random measure," Stochastic Processes and their Applications, Elsevier, vol. 123(2), pages 523-560.
    10. Pang, Guodong & Zhou, Yuhang, 2017. "Two-parameter process limits for an infinite-server queue with arrival dependent service times," Stochastic Processes and their Applications, Elsevier, vol. 127(5), pages 1375-1416.
    11. Torrisi, G. L., 2004. "Simulating the ruin probability of risk processes with delay in claim settlement," Stochastic Processes and their Applications, Elsevier, vol. 112(2), pages 225-244, August.
    12. Macci, Claudio & Torrisi, Giovanni Luca, 2004. "Asymptotic results for perturbed risk processes with delayed claims," Insurance: Mathematics and Economics, Elsevier, vol. 34(2), pages 307-320, April.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Chang Feng & John J. Hasenbein & Guodong Pang, 2025. "Sample-path moderate deviation principle for GI/GI/1+GI queues in the nearly critically loaded regime," Queueing Systems: Theory and Applications, Springer, vol. 109(2), pages 1-37, June.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Pang, Guodong & Zhou, Yuhang, 2018. "Functional limit theorems for a new class of non-stationary shot noise processes," Stochastic Processes and their Applications, Elsevier, vol. 128(2), pages 505-544.
    2. Martin Friesen & Stefan Gerhold & Kristof Wiedermann, 2024. "Small-time central limit theorems for stochastic Volterra integral equations and their Markovian lifts," Papers 2412.15971, arXiv.org.
    3. Marc Geha & Antoine Jacquier & Žan Žurič, 2024. "Large and moderate deviations for importance sampling in the Heston model," Annals of Operations Research, Springer, vol. 336(1), pages 47-92, May.
    4. Peter K. Friz & Thomas Wagenhofer, 2023. "Reconstructing volatility: Pricing of index options under rough volatility," Mathematical Finance, Wiley Blackwell, vol. 33(1), pages 19-40, January.
    5. Stabile, Gabriele & Torrisi, Giovanni Luca, 2010. "Large deviations of Poisson shot noise processes under heavy tail semi-exponential conditions," Statistics & Probability Letters, Elsevier, vol. 80(15-16), pages 1200-1209, August.
    6. Peter K. Friz & Thomas Wagenhofer, 2022. "Reconstructing Volatility: Pricing of Index Options under Rough Volatility," Papers 2212.07817, arXiv.org.
    7. Giacomo Giorgio & Barbara Pacchiarotti & Paolo Pigato, 2023. "Short-Time Asymptotics for Non-Self-Similar Stochastic Volatility Models," Applied Mathematical Finance, Taylor & Francis Journals, vol. 30(3), pages 123-152, May.
    8. Torrisi, Giovanni Luca & Leonardi, Emilio, 2022. "Asymptotic analysis of Poisson shot noise processes, and applications," Stochastic Processes and their Applications, Elsevier, vol. 144(C), pages 229-270.
    9. Maroulas, Vasileios & Pan, Xiaoyang & Xiong, Jie, 2020. "Large deviations for the optimal filter of nonlinear dynamical systems driven by Lévy noise," Stochastic Processes and their Applications, Elsevier, vol. 130(1), pages 203-231.
    10. Archil Gulisashvili, 2022. "Multivariate Stochastic Volatility Models and Large Deviation Principles," Papers 2203.09015, arXiv.org, revised Nov 2022.
    11. Dassios, Angelos & Jang, Jiwook & Zhao, Hongbiao, 2015. "A risk model with renewal shot-noise Cox process," Insurance: Mathematics and Economics, Elsevier, vol. 65(C), pages 55-65.
    12. Eduardo Abi Jaber & Shaun & Li, 2024. "Volatility models in practice: Rough, Path-dependent or Markovian?," Papers 2401.03345, arXiv.org, revised Apr 2025.
    13. Jiang Hui & Xu Lihu & Yang Qingshan, 2024. "Functional Large Deviations for Kac–Stroock Approximation to a Class of Gaussian Processes with Application to Small Noise Diffusions," Journal of Theoretical Probability, Springer, vol. 37(4), pages 3015-3054, November.
    14. J. G. Dai & Tolga Tezcan, 2011. "State Space Collapse in Many-Server Diffusion Limits of Parallel Server Systems," Mathematics of Operations Research, INFORMS, vol. 36(2), pages 271-320, May.
    15. Ioannis Dimitriou, 2025. "On Markov-dependent reflected autoregressive processes and related models," Queueing Systems: Theory and Applications, Springer, vol. 109(3), pages 1-39, September.
    16. Ma, Xiaocui & Xi, Fubao, 2017. "Moderate deviations for neutral stochastic differential delay equations with jumps," Statistics & Probability Letters, Elsevier, vol. 126(C), pages 97-107.
    17. Ruinan Li & Ran Wang & Beibei Zhang, 2024. "A Large Deviation Principle for the Stochastic Heat Equation with General Rough Noise," Journal of Theoretical Probability, Springer, vol. 37(1), pages 251-306, March.
    18. Deugoué, G. & Tachim Medjo, T., 2023. "Large deviation for a 3D globally modified Cahn–Hilliard–Navier–Stokes model under random influences," Stochastic Processes and their Applications, Elsevier, vol. 160(C), pages 33-71.
    19. Iksanov, Alexander, 2013. "Functional limit theorems for renewal shot noise processes with increasing response functions," Stochastic Processes and their Applications, Elsevier, vol. 123(6), pages 1987-2010.
    20. Ganguly, Arnab, 2018. "Large deviation principle for stochastic integrals and stochastic differential equations driven by infinite-dimensional semimartingales," Stochastic Processes and their Applications, Elsevier, vol. 128(7), pages 2179-2227.

    More about this item

    Keywords

    ;
    ;
    ;
    ;

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:spapps:v:176:y:2024:i:c:s0304414924001388. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/505572/description#description .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.