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A numerical algorithm for a class of BSDEs via the branching process

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  • Henry-Labordère, Pierre
  • Tan, Xiaolu
  • Touzi, Nizar

Abstract

We give a study to the algorithm for semi-linear parabolic PDEs in Henry-Labordère (2012) and then generalize it to the non-Markovian case for a class of Backward SDEs (BSDEs). By simulating the branching process, the algorithm does not need any backward regression. To prove that the numerical algorithm converges to the solution of BSDEs, we use the notion of viscosity solution of path dependent PDEs introduced by Ekren et al. (to appear) [5] and extended in Ekren et al. (2012) [6,7].

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:124:y:2014:i:2:p:1112-1140
    DOI: 10.1016/j.spa.2013.10.005
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    References listed on IDEAS

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    1. Rasulov, A. & Raimova, G. & Mascagni, M., 2010. "Monte Carlo solution of Cauchy problem for a nonlinear parabolic equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 80(6), pages 1118-1123.
    2. Nutz, Marcel & van Handel, Ramon, 2013. "Constructing sublinear expectations on path space," Stochastic Processes and their Applications, Elsevier, vol. 123(8), pages 3100-3121.
    3. 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.
    4. Marcel Nutz & Ramon van Handel, 2012. "Constructing Sublinear Expectations on Path Space," Papers 1205.2415, arXiv.org, revised Apr 2013.
    5. N. El Karoui & S. Peng & M. C. Quenez, 1997. "Backward Stochastic Differential Equations in Finance," Mathematical Finance, Wiley Blackwell, vol. 7(1), pages 1-71, January.
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    Cited by:

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    2. Christian Beck & Lukas Gonon & Arnulf Jentzen, 2024. "Overcoming the curse of dimensionality in the numerical approximation of high-dimensional semilinear elliptic partial differential equations," Partial Differential Equations and Applications, Springer, vol. 5(6), pages 1-47, December.
    3. Christoph Belak & Daniel Hoffmann & Frank T. Seifried, 2020. "Branching Diffusions with Jumps and Valuation with Systemic Counterparties," Working Paper Series 2020-04, University of Trier, Research Group Quantitative Finance and Risk Analysis.
    4. van der Zwaard, Thomas & Grzelak, Lech A. & Oosterlee, Cornelis W., 2021. "A computational approach to hedging Credit Valuation Adjustment in a jump-diffusion setting," Applied Mathematics and Computation, Elsevier, vol. 391(C).
    5. Antonis Papapantoleon & Dylan Possamai & Alexandros Saplaouras, 2021. "Stability of backward stochastic differential equations: the general case," Papers 2107.11048, arXiv.org, revised Apr 2023.
    6. 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.
    7. Pagès, Gilles & Sagna, Abass, 2018. "Improved error bounds for quantization based numerical schemes for BSDE and nonlinear filtering," Stochastic Processes and their Applications, Elsevier, vol. 128(3), pages 847-883.
    8. Agarwal, Ankush & Claisse, Julien, 2020. "Branching diffusion representation of semi-linear elliptic PDEs and estimation using Monte Carlo method," Stochastic Processes and their Applications, Elsevier, vol. 130(8), pages 5006-5036.
    9. Jean-Franc{c}ois Chassagneux & Junchao Chen & Noufel Frikha, 2022. "Deep Runge-Kutta schemes for BSDEs," Papers 2212.14372, arXiv.org.

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