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Representations and regularities for solutions to BSDEs with reflections

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  • Ma, Jin
  • Zhang, Jianfeng

Abstract

In this paper we study a class of backward stochastic differential equations with reflections (BSDER, for short). Three types of discretization procedures are introduced in the spirit of the so-called Bermuda Options in finance, so as to first establish a Feynman-Kac type formula for the martingale integrand of the BSDER, and then to derive the continuity of the paths of the martingale integrand, as well as the C1-regularity of the solution to a corresponding obstacle problem. We also introduce a new notion of regularity for a stochastic process, which we call the "L2-modulus regularity". Such a regularity is different from the usual path regularity in the literature, and we show that such regularity of the martingale integrand produces exactly the rate of convergence of a numerical scheme for BSDERs. Both numerical scheme and its rate of convergence are novel.

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  • Ma, Jin & Zhang, Jianfeng, 2005. "Representations and regularities for solutions to BSDEs with reflections," Stochastic Processes and their Applications, Elsevier, vol. 115(4), pages 539-569, April.
    • Handle: RePEc:eee:spapps:v:115:y:2005:i:4:p:539-569
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    References listed on IDEAS

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    1. Eric Fournié & Jean-Michel Lasry & Pierre-Louis Lions & Jérôme Lebuchoux & Nizar Touzi, 1999. "Applications of Malliavin calculus to Monte Carlo methods in finance," Finance and Stochastics, Springer, vol. 3(4), pages 391-412.
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    Cited by:

    1. Bouchard, Bruno & Chassagneux, Jean-François, 2008. "Discrete-time approximation for continuously and discretely reflected BSDEs," Stochastic Processes and their Applications, Elsevier, vol. 118(12), pages 2269-2293, December.
    2. Guangbao Guo, 2018. "Finite Difference Methods for the BSDEs in Finance," IJFS, MDPI, vol. 6(1), pages 1-15, March.
    3. Christian Bender & Nikolaus Schweizer & Jia Zhuo, 2013. "A primal-dual algorithm for BSDEs," Papers 1310.3694, arXiv.org, revised Sep 2014.
    4. Chassagneux, Jean-François & Richou, Adrien, 2019. "Rate of convergence for the discrete-time approximation of reflected BSDEs arising in switching problems," Stochastic Processes and their Applications, Elsevier, vol. 129(11), pages 4597-4637.
    5. Erhan Bayraktar & Qi Feng & Zhaoyu Zhang, 2022. "Deep Signature Algorithm for Multi-dimensional Path-Dependent Options," Papers 2211.11691, arXiv.org, revised Jan 2024.
    6. Fahrenwaldt, Matthias A. & Sun, Chaofan, 2020. "Expected utility approximation and portfolio optimisation," Insurance: Mathematics and Economics, Elsevier, vol. 93(C), pages 301-314.
    7. Erhan Bayraktar & Arash Fahim, 2011. "A Stochastic Approximation for Fully Nonlinear Free Boundary Parabolic Problems," Papers 1109.5752, arXiv.org, revised Nov 2013.
    8. Teng, Long, 2022. "Gradient boosting-based numerical methods for high-dimensional backward stochastic differential equations," Applied Mathematics and Computation, Elsevier, vol. 426(C).
    9. Dingqian Sun, 2020. "The convergence rate from discrete to continuous optimal investment stopping problem," Papers 2004.14627, arXiv.org.
    10. Benedetti, Giuseppe & Campi, Luciano, 2016. "Utility indifference valuation for non-smooth payoffs with an application to power derivatives," LSE Research Online Documents on Economics 63016, London School of Economics and Political Science, LSE Library.

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