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A finite-dimensional approximation for partial differential equations on Wasserstein space

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  • Talbi, Mehdi

Abstract

This paper presents a finite-dimensional approximation for a class of partial differential equations on the space of probability measures. These equations are satisfied in the sense of viscosity solutions. The main result states the convergence of the viscosity solutions of the finite-dimensional PDE to the viscosity solutions of the PDE on Wasserstein space, provided that uniqueness holds for the latter, and heavily relies on an adaptation of the Barles & Souganidis monotone scheme (Barles and Souganidis, 1991) to our context, as well as on a key precompactness result for semimartingale measures. We illustrate our convergence result with the example of the Hamilton–Jacobi–Bellman and Bellman–Isaacs equations arising in stochastic control and differential games, and propose an extension to the case of path-dependent PDEs.

Suggested Citation

  • Talbi, Mehdi, 2024. "A finite-dimensional approximation for partial differential equations on Wasserstein space," Stochastic Processes and their Applications, Elsevier, vol. 177(C).
  • Handle: RePEc:eee:spapps:v:177:y:2024:i:c:s0304414924001510
    DOI: 10.1016/j.spa.2024.104445
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    References listed on IDEAS

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    1. Jianfeng Zhang & Jia Zhuo, 2014. "Monotone schemes for fully nonlinear parabolic path dependent PDEs," Journal of Financial Engineering (JFE), World Scientific Publishing Co. Pte. Ltd., vol. 1(01), pages 1-23.
    2. Cecchin, Alekos & Pelino, Guglielmo, 2019. "Convergence, fluctuations and large deviations for finite state mean field games via the Master Equation," Stochastic Processes and their Applications, Elsevier, vol. 129(11), pages 4510-4555.
    3. Lacker, Daniel, 2015. "Mean field games via controlled martingale problems: Existence of Markovian equilibria," Stochastic Processes and their Applications, Elsevier, vol. 125(7), pages 2856-2894.
    4. Ren, Zhenjie & Tan, Xiaolu, 2017. "On the convergence of monotone schemes for path-dependent PDEs," Stochastic Processes and their Applications, Elsevier, vol. 127(6), pages 1738-1762.
    5. Maximilien Germain & Huyên Pham & Xavier Warin, 2022. "Rate of convergence for particle approximation of PDEs in Wasserstein space ," Post-Print hal-03154021, HAL.
    6. Bruno Dupire, 2019. "Functional Itô calculus," Quantitative Finance, Taylor & Francis Journals, vol. 19(5), pages 721-729, May.
    7. Justin Sirignano & Konstantinos Spiliopoulos, 2017. "DGM: A deep learning algorithm for solving partial differential equations," Papers 1708.07469, arXiv.org, revised Sep 2018.
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