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PDGM: a Neural Network Approach to Solve Path-Dependent Partial Differential Equations

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  • Yuri F. Saporito
  • Zhaoyu Zhang

Abstract

In this paper, we propose a novel numerical method for Path-Dependent Partial Differential Equations (PPDEs). These equations firstly appeared in the seminal work of Dupire [2009], where the functional It\^o calculus was developed to deal with path-dependent financial derivatives contracts. More specificaly, we generalize the Deep Galerking Method (DGM) of Sirignano and Spiliopoulos [2018] to deal with these equations. The method, which we call Path-Dependent DGM (PDGM), consists of using a combination of feed-forward and Long Short-Term Memory architectures to model the solution of the PPDE. We then analyze several numerical examples, many from the Financial Mathematics literature, that show the capabilities of the method under very different situations.

Suggested Citation

  • 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.
    • Handle: RePEc:arx:papers:2003.02035
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    References listed on IDEAS

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    1. 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.
    2. Samy Jazaerli & Yuri F. Saporito, 2013. "Functional Ito Calculus, Path-dependence and the Computation of Greeks," Papers 1311.3881, arXiv.org, revised Jun 2018.
    3. Justin Sirignano & Konstantinos Spiliopoulos, 2017. "DGM: A deep learning algorithm for solving partial differential equations," Papers 1708.07469, arXiv.org, revised Sep 2018.
    4. Emmanuel Gobet, 2009. "Advanced Monte Carlo methods for barrier and related exotic options," Post-Print hal-00319947, HAL.
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    Cited by:

    1. Marc Sabate-Vidales & David v{S}iv{s}ka & Lukasz Szpruch, 2020. "Solving path dependent PDEs with LSTM networks and path signatures," Papers 2011.10630, arXiv.org.
    2. Zhou Fang, 2023. "Continuous-Time Path-Dependent Exploratory Mean-Variance Portfolio Construction," Papers 2303.02298, arXiv.org.
    3. Qi Feng & Man Luo & Zhaoyu Zhang, 2021. "Deep Signature FBSDE Algorithm," Papers 2108.10504, arXiv.org, revised Aug 2022.
    4. 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.
    5. Sebastian Jaimungal, 2022. "Reinforcement learning and stochastic optimisation," Finance and Stochastics, Springer, vol. 26(1), pages 103-129, January.

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