IDEAS home Printed from https://ideas.repec.org/a/spr/metcap/v24y2022i3d10.1007_s11009-021-09871-9.html
   My bibliography  Save this article

Solving Elliptic Equations with Brownian Motion: Bias Reduction and Temporal Difference Learning

Author

Listed:
  • Cameron Martin

    (University of Toronto)

  • Hongyuan Zhang

    (Carnegie Mellon University)

  • Julia Costacurta

    (Stanford University)

  • Mihai Nica

    (University of Guelph)

  • Adam R Stinchcombe

    (University of Toronto)

Abstract

The Feynman-Kac formula provides a way to understand solutions to elliptic partial differential equations in terms of expectations of continuous time Markov processes. This connection allows for the creation of numerical schemes for solutions based on samples of these Markov processes which have advantages over traditional numerical methods in some cases. However, naïve numerical implementations suffer from issues related to statistical bias and sampling efficiency. We present methods to discretize the stochastic process appearing in the Feynman-Kac formula that reduce the bias of the numerical scheme. We also propose using temporal difference learning to assemble information from random samples in a way that is more efficient than the traditional Monte Carlo method.

Suggested Citation

  • Cameron Martin & Hongyuan Zhang & Julia Costacurta & Mihai Nica & Adam R Stinchcombe, 2022. "Solving Elliptic Equations with Brownian Motion: Bias Reduction and Temporal Difference Learning," Methodology and Computing in Applied Probability, Springer, vol. 24(3), pages 1603-1626, September.
    • Handle: RePEc:spr:metcap:v:24:y:2022:i:3:d:10.1007_s11009-021-09871-9
    DOI: 10.1007/s11009-021-09871-9
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s11009-021-09871-9
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s11009-021-09871-9?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    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. Lejay, Antoine & Maire, Sylvain, 2007. "Computing the principal eigenvalue of the Laplace operator by a stochastic method," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 73(6), pages 351-363.
    2. Mark Broadie & Paul Glasserman & Steven Kou, 1997. "A Continuity Correction for Discrete Barrier Options," Mathematical Finance, Wiley Blackwell, vol. 7(4), pages 325-349, October.
    3. Hwang, Chi-Ok & Mascagni, Michael & Given, James A., 2003. "A Feynman–Kac path-integral implementation for Poisson’s equation using an h-conditioned Green’s function," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 62(3), pages 347-355.
    4. Gobet, Emmanuel & Menozzi, Stéphane, 2010. "Stopped diffusion processes: Boundary corrections and overshoot," Stochastic Processes and their Applications, Elsevier, vol. 120(2), pages 130-162, February.
    5. Justin Sirignano & Konstantinos Spiliopoulos, 2017. "DGM: A deep learning algorithm for solving partial differential equations," Papers 1708.07469, arXiv.org, revised Sep 2018.
    Full references (including those not matched with items on IDEAS)

    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. Herrmann, Samuel & Massin, Nicolas, 2023. "Exact simulation of the first passage time through a given level of jump diffusions," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 203(C), pages 553-576.
    2. Sheng-Feng Luo & Hsin-Chieh Wong, 2023. "Continuity correction: on the pricing of discrete double barrier options," Review of Derivatives Research, Springer, vol. 26(1), pages 51-90, April.
    3. William Lefebvre & Enzo Miller, 2021. "Linear-quadratic stochastic delayed control and deep learning resolution," Working Papers hal-03145949, HAL.
    4. A. Max Reppen & H. Mete Soner & Valentin Tissot-Daguette, 2022. "Deep Stochastic Optimization in Finance," Papers 2205.04604, arXiv.org.
    5. Sebastian Jaimungal, 2022. "Reinforcement learning and stochastic optimisation," Finance and Stochastics, Springer, vol. 26(1), pages 103-129, January.
    6. Xie, Fei & He, Zhijian & Wang, Xiaoqun, 2019. "An importance sampling-based smoothing approach for quasi-Monte Carlo simulation of discrete barrier options," European Journal of Operational Research, Elsevier, vol. 274(2), pages 759-772.
    7. N. Hilber & N. Reich & C. Schwab & C. Winter, 2009. "Numerical methods for Lévy processes," Finance and Stochastics, Springer, vol. 13(4), pages 471-500, September.
    8. Kontosakos, Vasileios E. & Mendonca, Keegan & Pantelous, Athanasios A. & Zuev, Konstantin M., 2021. "Pricing discretely-monitored double barrier options with small probabilities of execution," European Journal of Operational Research, Elsevier, vol. 290(1), pages 313-330.
    9. Jorge Ignacio Gonz'alez C'azares & Aleksandar Mijatovi'c & Ger'onimo Uribe Bravo, 2018. "Geometrically Convergent Simulation of the Extrema of L\'{e}vy Processes," Papers 1810.11039, arXiv.org, revised Jun 2021.
    10. Rong Du & Duy-Minh Dang, 2023. "Fourier Neural Network Approximation of Transition Densities in Finance," Papers 2309.03966, arXiv.org, revised Sep 2024.
    11. Gero Junike & Hauke Stier, 2024. "Enhancing Fourier pricing with machine learning," Papers 2412.05070, arXiv.org.
    12. Ali Al-Aradi & Adolfo Correia & Danilo de Frietas Naiff & Gabriel Jardim & Yuri Saporito, 2019. "Extensions of the Deep Galerkin Method," Papers 1912.01455, arXiv.org, revised Apr 2022.
    13. Lokeshwar, Vikranth & Bharadwaj, Vikram & Jain, Shashi, 2022. "Explainable neural network for pricing and universal static hedging of contingent claims," Applied Mathematics and Computation, Elsevier, vol. 417(C).
    14. Salah A. Faroughi & Ramin Soltanmohammadi & Pingki Datta & Seyed Kourosh Mahjour & Shirko Faroughi, 2023. "Physics-Informed Neural Networks with Periodic Activation Functions for Solute Transport in Heterogeneous Porous Media," Mathematics, MDPI, vol. 12(1), pages 1-23, December.
    15. Jiequn Han & Ruimeng Hu & Jihao Long, 2020. "Convergence of Deep Fictitious Play for Stochastic Differential Games," Papers 2008.05519, arXiv.org, revised Mar 2021.
    16. C. E. Phelan & D. Marazzina & G. Germano, 2020. "Pricing methods for α-quantile and perpetual early exercise options based on Spitzer identities," Quantitative Finance, Taylor & Francis Journals, vol. 20(6), pages 899-918, June.
    17. Tian-Shyr Dai & Yuh-Yuan Fang & Yuh-Dauh Lyuu, 2005. "Analytics for geometric average trigger reset options," Applied Economics Letters, Taylor & Francis Journals, vol. 12(13), pages 835-840.
    18. Dehghani, Hamidreza & Zilian, Andreas, 2021. "A hybrid MGA-MSGD ANN training approach for approximate solution of linear elliptic PDEs," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 190(C), pages 398-417.
    19. Ernst, Philip A. & Rogers, L.C.G. & Zhou, Quan, 2017. "The value of foresight," Stochastic Processes and their Applications, Elsevier, vol. 127(12), pages 3913-3927.
    20. Jesús Fernández-Villaverde & Galo Nuño & Jesse Perla, 2024. "Taming the Curse of Dimensionality: Quantitative Economics with Deep Learning," NBER Working Papers 33117, National Bureau of Economic Research, Inc.

    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:spr:metcap:v:24:y:2022:i:3:d:10.1007_s11009-021-09871-9. 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: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    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.