IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v83y1999i1p103-126.html
   My bibliography  Save this article

Particle representations for a class of nonlinear SPDEs

Author

Listed:
  • Kurtz, Thomas G.
  • Xiong, Jie

Abstract

An infinite system of stochastic differential equations for the locations and weights of a collection of particles is considered. The particles interact through their weighted empirical measure, V, and V is shown to be the unique solution of a nonlinear stochastic partial differential equation (SPDE). Conditions are given under which the weighted empirical measure has an L2-density with respect to Lebesgue measure.

Suggested Citation

  • Kurtz, Thomas G. & Xiong, Jie, 1999. "Particle representations for a class of nonlinear SPDEs," Stochastic Processes and their Applications, Elsevier, vol. 83(1), pages 103-126, September.
    • Handle: RePEc:eee:spapps:v:83:y:1999:i:1:p:103-126
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304-4149(99)00024-1
    Download Restriction: Full text for ScienceDirect subscribers only
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Hitsuda, Masuyuki & Mitoma, Itaru, 1986. "Tightness problem and stochastic evolution equation arising from fluctuation phenomena for interacting diffusions," Journal of Multivariate Analysis, Elsevier, vol. 19(2), pages 311-328, August.
    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. Budhiraja, Amarjit & Wu, Ruoyu, 2016. "Some fluctuation results for weakly interacting multi-type particle systems," Stochastic Processes and their Applications, Elsevier, vol. 126(8), pages 2253-2296.
    2. Ben Hambly & Nikolaos Kolliopoulos, 2018. "Fast mean-reversion asymptotics for large portfolios of stochastic volatility models," Papers 1811.08808, arXiv.org, revised Feb 2020.
    3. Christa Cuchiero & Martin Larsson & Sara Svaluto-Ferro, 2018. "Probability measure-valued polynomial diffusions," Papers 1807.03229, arXiv.org.
    4. Bayraktar, Erhan & Wu, Ruoyu, 2021. "Mean field interaction on random graphs with dynamically changing multi-color edges," Stochastic Processes and their Applications, Elsevier, vol. 141(C), pages 197-244.
    5. Ben Hambly & Nikolaos Kolliopoulos, 2019. "Stochastic PDEs for large portfolios with general mean-reverting volatility processes," Papers 1906.05898, arXiv.org, revised Mar 2024.
    6. Ahmad, F. & Hambly, B.M. & Ledger, S., 2018. "A stochastic partial differential equation model for the pricing of mortgage-backed securities," Stochastic Processes and their Applications, Elsevier, vol. 128(11), pages 3778-3806.
    7. Amarjit Budhiraja & Michael Conroy, 2022. "Empirical Measure and Small Noise Asymptotics Under Large Deviation Scaling for Interacting Diffusions," Journal of Theoretical Probability, Springer, vol. 35(1), pages 295-349, March.
    8. 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.
    9. Jie Xiong & Yong Zeng, 2011. "A branching particle approximation to a filtering micromovement model of asset price," Statistical Inference for Stochastic Processes, Springer, vol. 14(2), pages 111-140, May.
    10. Nguyen, Son L. & Yin, George & Hoang, Tuan A., 2020. "On laws of large numbers for systems with mean-field interactions and Markovian switching," Stochastic Processes and their Applications, Elsevier, vol. 130(1), pages 262-296.
    11. Calvia, Alessandro & Ferrari, Giorgio, 2021. "Nonlinear Filtering of Partially Observed Systems Arising in Singular Stochastic Optimal Control," Center for Mathematical Economics Working Papers 651, Center for Mathematical Economics, Bielefeld University.
    12. Lijun Bo & Tongqing Li & Xiang Yu, 2021. "Centralized systemic risk control in the interbank system: Weak formulation and Gamma-convergence," Papers 2106.09978, arXiv.org, revised May 2022.
    13. Rémillard, Bruno & Vaillancourt, Jean, 2014. "On signed measure valued solutions of stochastic evolution equations," Stochastic Processes and their Applications, Elsevier, vol. 124(1), pages 101-122.
    14. Clini, Andrea, 2023. "Porous media equations with nonlinear gradient noise and Dirichlet boundary conditions," Stochastic Processes and their Applications, Elsevier, vol. 159(C), pages 428-498.
    15. Coghi, Michele & Nilssen, Torstein, 2021. "Rough nonlocal diffusions," Stochastic Processes and their Applications, Elsevier, vol. 141(C), pages 1-56.
    16. Michael B. Giles & Christoph Reisinger, 2012. "Stochastic finite differences and multilevel Monte Carlo for a class of SPDEs in finance," Papers 1204.1442, arXiv.org.
    17. Rene Carmona & Kevin Webster, 2012. "High Frequency Market Making," Papers 1210.5781, arXiv.org.
    18. Josselin Garnier & George Papanicolaou & Tzu-Wei Yang, 2015. "A risk analysis for a system stabilized by a central agent," Papers 1507.08333, arXiv.org, revised Aug 2015.
    19. Matthieu Gomez, 2023. "Decomposing the Growth of Top Wealth Shares," Econometrica, Econometric Society, vol. 91(3), pages 979-1024, May.
    20. Bo, Lijun & Li, Tongqing & Yu, Xiang, 2022. "Centralized systemic risk control in the interbank system: Weak formulation and Gamma-convergence," Stochastic Processes and their Applications, Elsevier, vol. 150(C), pages 622-654.
    21. Bhamidi, Shankar & Budhiraja, Amarjit & Wu, Ruoyu, 2019. "Weakly interacting particle systems on inhomogeneous random graphs," Stochastic Processes and their Applications, Elsevier, vol. 129(6), pages 2174-2206.

    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. Chevallier, Julien & Ost, Guilherme, 2020. "Fluctuations for spatially extended Hawkes processes," Stochastic Processes and their Applications, Elsevier, vol. 130(9), pages 5510-5542.
    2. Fernandez, Begoña & Méléard, Sylvie, 1997. "A Hilbertian approach for fluctuations on the McKean-Vlasov model," Stochastic Processes and their Applications, Elsevier, vol. 71(1), pages 33-53, October.
    3. Jie Xiong & Yong Zeng, 2011. "A branching particle approximation to a filtering micromovement model of asset price," Statistical Inference for Stochastic Processes, Springer, vol. 14(2), pages 111-140, May.

    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:83:y:1999:i:1:p:103-126. 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.