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On the robustness of backward stochastic differential equations

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  • Briand, Philippe
  • Delyon, Bernard
  • Mémin, Jean

Abstract

In this paper, we study the robustness of backward stochastic differential equations (BSDEs for short) w.r.t. the Brownian motion; more precisely, we will show that if Wn is a martingale approximation of a Brownian motion W then the solution to the BSDE driven by the martingale Wn converges to the solution of the classical BSDE, namely the BSDE driven by W. The particular case of the scaled random walks has been studied in Briand et al. (Electron. Comm. Probab. 6 (2001) 1). Here, we deal with a more general situation and we will not assume that the Wn has the predictable representation property: this yields an orthogonal martingale in the BSDE driven by Wn. As a byproduct of our result, we obtain the convergence of the "Euler scheme" for BSDEs corresponding to the case where Wn is a time discretization of W.

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  • Briand, Philippe & Delyon, Bernard & Mémin, Jean, 2002. "On the robustness of backward stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 97(2), pages 229-253, February.
    • Handle: RePEc:eee:spapps:v:97:y:2002:i:2:p:229-253
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    References listed on IDEAS

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    1. 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.
    2. Jacod, J. & Memin, J. & Metivier, M., 1983. "On tightness and stopping times," Stochastic Processes and their Applications, Elsevier, vol. 14(2), pages 109-146, February.
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    Cited by:

    1. Geiss, Christel & Labart, Céline, 2016. "Simulation of BSDEs with jumps by Wiener Chaos expansion," Stochastic Processes and their Applications, Elsevier, vol. 126(7), pages 2123-2162.
    2. Ismail Laachir & Francesco Russo, 2016. "BSDEs, càdlàg martingale problems and orthogonalisation under basis risk," Working Papers hal-01086227, HAL.
    3. Ceci, Claudia & Cretarola, Alessandra & Russo, Francesco, 2014. "BSDEs under partial information and financial applications," Stochastic Processes and their Applications, Elsevier, vol. 124(8), pages 2628-2653.
    4. Christoph Czichowsky, 2012. "Time-Consistent Mean-Variance Portfolio Selection in Discrete and Continuous Time," Papers 1205.4748, arXiv.org.
    5. Blessing, Jonas & Kupper, Michael & Nendel, Max, 2023. "Convergence of Infintesimal Generators and Stability of Convex Montone Semigroups," Center for Mathematical Economics Working Papers 680, Center for Mathematical Economics, Bielefeld University.
    6. Bender, Christian & Parczewski, Peter, 2018. "Discretizing Malliavin calculus," Stochastic Processes and their Applications, Elsevier, vol. 128(8), pages 2489-2537.
    7. Yao, Song, 2017. "Lp solutions of backward stochastic differential equations with jumps," Stochastic Processes and their Applications, Elsevier, vol. 127(11), pages 3465-3511.
    8. Madan, Dilip & Pistorius, Martijn & Stadje, Mitja, 2016. "Convergence of BSΔEs driven by random walks to BSDEs: The case of (in)finite activity jumps with general driver," Stochastic Processes and their Applications, Elsevier, vol. 126(5), pages 1553-1584.
    9. Christoph Czichowsky, 2013. "Time-consistent mean-variance portfolio selection in discrete and continuous time," Finance and Stochastics, Springer, vol. 17(2), pages 227-271, April.
    10. Stadje, Mitja, 2010. "Extending dynamic convex risk measures from discrete time to continuous time: A convergence approach," Insurance: Mathematics and Economics, Elsevier, vol. 47(3), pages 391-404, December.
    11. Cheridito, Patrick & Stadje, Mitja, 2012. "Existence, minimality and approximation of solutions to BSDEs with convex drivers," Stochastic Processes and their Applications, Elsevier, vol. 122(4), pages 1540-1565.
    12. Zhang, Guichang, 2006. "Discretization of backward semilinear stochastic evolution equations," Stochastic Processes and their Applications, Elsevier, vol. 116(8), pages 1097-1126, August.
    13. 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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