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Entropy-regularized penalization schemes and reflected BSDEs with singular generators

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  • Daniel Chee
  • Noufel Frikha
  • Libo Li

Abstract

This paper extends our previous work to continuous-time optimal stopping, focusing on American options in an exploratory setting. Our first contribution is an entropy-regularized penalization scheme, inspired by classical penalization techniques for reflected BSDEs. It yields a smooth approximation of the stopping rule, promotes exploration, and enables gradient-based learning methods. We prove well-posedness, convergence, and illustrate numerical performance in low-dimensional examples. Our second contribution analyzes the behaviour of the scheme as the penalization parameter grows, showing that the limit solves a reflected BSDE with a logarithmically singular generator, for which we establish existence and uniqueness via a monotone limit argument.

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  • Daniel Chee & Noufel Frikha & Libo Li, 2026. "Entropy-regularized penalization schemes and reflected BSDEs with singular generators," Papers 2602.18078, arXiv.org, revised Mar 2026.
    • Handle: RePEc:arx:papers:2602.18078
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    References listed on IDEAS

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    1. Leif Andersen & Mark Broadie, 2004. "Primal-Dual Simulation Algorithm for Pricing Multidimensional American Options," Management Science, INFORMS, vol. 50(9), pages 1222-1234, September.
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    3. Miryana Grigorova & Peter Imkeller & Elias Offen & Youssef Ouknine & Marie-Claire Quenez, 2017. "Reflected BSDEs when the obstacle is not right-continuous and optimal stopping," Post-Print hal-01141801, HAL.
    4. Daniel Chee & Noufel Frikha & Libo Li, 2026. "A Monotone Limit Approach to Entropy-Regularized American Options," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) hal-05520656, HAL.
    5. Zhen-Qing Chen & Xinwei Feng, 2021. "Reflected Backward Stochastic Differential Equation with Rank-Based Data," Journal of Theoretical Probability, Springer, vol. 34(3), pages 1213-1247, September.
    6. Miryana Grigorova & Peter Imkeller & Elias Offen & Youssef Ouknine & Marie-Claire Quenez, 2015. "Reflected BSDEs when the obstacle is not right-continuous and optimal stopping," Papers 1504.06094, arXiv.org, revised May 2017.
    7. Duffie, Darrell & Epstein, Larry G, 1992. "Stochastic Differential Utility," Econometrica, Econometric Society, vol. 60(2), pages 353-394, March.
    8. Wenbo Wang & Guangyan Jia, 2025. "Quadratic BSDEs with Singular Generators and Unbounded Terminal Conditions: Theory and Applications," Mathematics, MDPI, vol. 13(14), pages 1-27, July.
    9. Duffie, Darrell & Epstein, Larry G, 1992. "Asset Pricing with Stochastic Differential Utility," The Review of Financial Studies, Society for Financial Studies, vol. 5(3), pages 411-436.
    10. Daniel Chee & Noufel Frikha & Libo Li, 2026. "A Monotone Limit Approach to Entropy-Regularized American Options," Papers 2602.18062, arXiv.org.
    11. Noufel Frikha & Libo Li & Daniel Chee, 2025. "An Entropy Regularized BSDE Approach to Bermudan Options and Games," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) hal-05265653, HAL.
    12. Andres Max Reppen & Halil Mete Soner & Valentin Tissot‐Daguette, 2025. "Neural optimal stopping boundary," Mathematical Finance, Wiley Blackwell, vol. 35(2), pages 441-469, April.
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