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On the convergence of monotone schemes for path-dependent PDEs

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  • Ren, Zhenjie
  • Tan, Xiaolu

Abstract

We propose a reformulation of the convergence theorem of monotone numerical schemes introduced by Zhang and Zhuo (2014) for viscosity solutions to path-dependent PDEs (PPDE), which extends the seminal work of Barles and Souganidis (1991) on the viscosity solution to PDE. We prove the convergence theorem under conditions similar to those of the classical theorem in Barles and Souganidis (1991). These conditions are satisfied, to the best of our knowledge, by all classical monotone numerical schemes in the context of stochastic control theory. In particular, the paper provides a unified approach to prove the convergence of numerical schemes for non-Markovian stochastic control problems, second order BSDEs, stochastic differential games, etc.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:127:y:2017:i:6:p:1738-1762
    DOI: 10.1016/j.spa.2016.10.002
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    References listed on IDEAS

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    1. Henry-Labordère, Pierre & Tan, Xiaolu & Touzi, Nizar, 2014. "A numerical algorithm for a class of BSDEs via the branching process," Stochastic Processes and their Applications, Elsevier, vol. 124(2), pages 1112-1140.
    2. repec:dau:papers:123456789/5524 is not listed on IDEAS
    3. 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.
    4. Bouchard, Bruno & Touzi, Nizar, 2004. "Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 111(2), pages 175-206, June.
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    Citations

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    Cited by:

    1. Thibaut Mastrolia & Dylan Possamaï, 2018. "Moral Hazard Under Ambiguity," Journal of Optimization Theory and Applications, Springer, vol. 179(2), pages 452-500, November.
    2. Jun Moon, 2022. "State and Control Path-Dependent Stochastic Zero-Sum Differential Games: Viscosity Solutions of Path-Dependent Hamilton–Jacobi–Isaacs Equations," Mathematics, MDPI, vol. 10(10), pages 1-32, May.
    3. Jiang Yu Nguwi & Nicolas Privault, 2023. "A deep learning approach to the probabilistic numerical solution of path-dependent partial differential equations," Partial Differential Equations and Applications, Springer, vol. 4(4), pages 1-20, August.
    4. Bouchard, Bruno & Loeper, Grégoire & Tan, Xiaolu, 2022. "A ℂ0,1-functional Itô’s formula and its applications in mathematical finance," Stochastic Processes and their Applications, Elsevier, vol. 148(C), pages 299-323.
    5. Yuri F. Saporito & Zhaoyu Zhang, 2020. "PDGM: a Neural Network Approach to Solve Path-Dependent Partial Differential Equations," Papers 2003.02035, arXiv.org, revised Apr 2020.
    6. Bruno Bouchard & Grégoire Loeper & Xiaolu Tan, 2022. "A C^{0,1}-functional Itô's formula and its applications in mathematical finance," Post-Print hal-03105342, HAL.

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