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Improved error bounds for quantization based numerical schemes for BSDE and nonlinear filtering

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  • Pagès, Gilles
  • Sagna, Abass

Abstract

We take advantage of recent (see Graf et al., 2008; Pages and Wilbertz, 2012) and new results on optimal quantization theory to improve the quadratic optimal quantization error bounds for backward stochastic differential equations (BSDE) and nonlinear filtering problems. For both problems, a first improvement relies on a Pythagoras like Theorem for quantized conditional expectation. While allowing for some locally Lipschitz continuous conditional densities in nonlinear filtering, the analysis of the error brings into play a new robustness result about optimal quantizers, the so-called distortion mismatch property: the Ls-mean quantization error induced by Lr-optimal quantizers of size N converges at the same rate N−1d for every s∈(0,r+d).

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  • 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.
    • Handle: RePEc:eee:spapps:v:128:y:2018:i:3:p:847-883
    DOI: 10.1016/j.spa.2017.05.009
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    References listed on IDEAS

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    1. Bally, Vlad & Pagès, Gilles, 2003. "Error analysis of the optimal quantization algorithm for obstacle problems," Stochastic Processes and their Applications, Elsevier, vol. 106(1), pages 1-40, July.
    2. 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.
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    4. Bender, Christian & Denk, Robert, 2007. "A forward scheme for backward SDEs," Stochastic Processes and their Applications, Elsevier, vol. 117(12), pages 1793-1812, December.
    5. Vlad Bally & Gilles Pagès & Jacques Printems, 2005. "A Quantization Tree Method For Pricing And Hedging Multidimensional American Options," Mathematical Finance, Wiley Blackwell, vol. 15(1), pages 119-168, January.
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    7. repec:dau:papers:123456789/5522 is not listed on IDEAS
    8. 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.
    9. 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.
    10. Bergman, Yaacov Z, 1995. "Option Pricing with Differential Interest Rates," Review of Financial Studies, Society for Financial Studies, vol. 8(2), pages 475-500.
    11. Geiss, Christel & Geiss, Stefan & Gobet, Emmanuel, 2012. "Generalized fractional smoothness and Lp-variation of BSDEs with non-Lipschitz terminal condition," Stochastic Processes and their Applications, Elsevier, vol. 122(5), pages 2078-2116.
    12. Anne Laure Bronstein & Gilles Pages & Benedikt Wilbertz, 2010. "How to speed up the quantization tree algorithm with an application to swing options," Quantitative Finance, Taylor & Francis Journals, vol. 10(9), pages 995-1007.
    13. Gobet, Emmanuel & Labart, Céline, 2007. "Error expansion for the discretization of backward stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 117(7), pages 803-829, July.
    14. Crisan, D. & Manolarakis, K. & Touzi, N., 2010. "On the Monte Carlo simulation of BSDEs: An improvement on the Malliavin weights," Stochastic Processes and their Applications, Elsevier, vol. 120(7), pages 1133-1158, July.
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    Citations

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    Cited by:

    1. Jean-Franc{c}ois Chassagneux & Junchao Chen & Noufel Frikha, 2022. "Deep Runge-Kutta schemes for BSDEs," Papers 2212.14372, arXiv.org.
    2. Giorgia Callegaro & Alessandro Gnoatto & Martino Grasselli, 2021. "A Fully Quantization-based Scheme for FBSDEs," Working Papers 07/2021, University of Verona, Department of Economics.
    3. Vincent Lemaire & Thibaut Montes & Gilles Pagès, 2019. "New Weak Error bounds and expansions for Optimal Quantization," Working Papers hal-02361644, HAL.
    4. Jean-Franc{c}ois Chassagneux & Mohan Yang, 2021. "Numerical approximation of singular Forward-Backward SDEs," Papers 2106.15496, arXiv.org.
    5. Lucio Fiorin & Gilles Pagès & Abass Sagna, 2019. "Product Markovian Quantization of a Diffusion Process with Applications to Finance," Methodology and Computing in Applied Probability, Springer, vol. 21(4), pages 1087-1118, December.
    6. Jean-Michel Fayolle & Vincent Lemaire & Thibaut Montes & Gilles Pagès, 2020. "Quantization-based Bermudan option pricing in the FX world," Working Papers hal-02361667, HAL.
    7. Vincent Lemaire & Thibaut Montes & Gilles Pagès, 2020. "New Weak Error bounds and expansions for Optimal Quantization," Post-Print hal-02361644, HAL.
    8. Antonis Papapantoleon & Dylan Possamai & Alexandros Saplaouras, 2021. "Stability of backward stochastic differential equations: the general case," Papers 2107.11048, arXiv.org, revised Apr 2023.
    9. Jean-Michel Fayolle & Vincent Lemaire & Thibaut Montes & Gilles Pag`es, 2019. "Quantization-based Bermudan option pricing in the $FX$ world," Papers 1911.05462, arXiv.org, revised May 2020.
    10. Callegaro, Giorgia & Gnoatto, Alessandro & Grasselli, Martino, 2023. "A fully quantization-based scheme for FBSDEs," Applied Mathematics and Computation, Elsevier, vol. 441(C).

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