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Particle representations for a class of nonlinear SPDEs

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  • 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
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    References listed on IDEAS

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    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.
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    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. 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.
    4. Coghi, Michele & Nilssen, Torstein, 2021. "Rough nonlocal diffusions," Stochastic Processes and their Applications, Elsevier, vol. 141(C), pages 1-56.
    5. 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.
    6. Christa Cuchiero & Martin Larsson & Sara Svaluto-Ferro, 2018. "Probability measure-valued polynomial diffusions," Papers 1807.03229, arXiv.org.
    7. Rene Carmona & Kevin Webster, 2012. "High Frequency Market Making," Papers 1210.5781, arXiv.org.
    8. 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.
    9. Ben Hambly & Nikolaos Kolliopoulos, 2019. "Stochastic PDEs for large portfolios with general mean-reverting volatility processes," Papers 1906.05898, arXiv.org, revised Mar 2024.
    10. 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.
    11. 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.
    12. 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.
    13. 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.
    14. 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.
    15. 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.
    16. Matthieu Gomez, 2023. "Decomposing the Growth of Top Wealth Shares," Econometrica, Econometric Society, vol. 91(3), pages 979-1024, May.
    17. 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.
    18. 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.
    19. 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.
    20. 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.
    21. 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.

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