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Existence, minimality and approximation of solutions to BSDEs with convex drivers

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  • Cheridito, Patrick
  • Stadje, Mitja

Abstract

We study the existence of solutions to backward stochastic differential equations with drivers f(t,W,y,z) that are convex in z. We assume f to be Lipschitz in y and W but do not make growth assumptions with respect to z. We first show the existence of a unique solution (Y,Z) with bounded Z if the terminal condition is Lipschitz in W and that it can be approximated by the solutions to properly discretized equations. If the terminal condition is bounded and uniformly continuous in W we show the existence of a minimal continuous supersolution by uniformly approximating the terminal condition with Lipschitz terminal conditions. Finally, we prove the existence of a minimal RCLL supersolution for bounded lower semicontinuous terminal conditions by approximating the terminal condition pointwise from below with Lipschitz terminal conditions.

Suggested Citation

  • 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.
  • Handle: RePEc:eee:spapps:v:122:y:2012:i:4:p:1540-1565
    DOI: 10.1016/j.spa.2011.12.008
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    References listed on IDEAS

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    1. Freddy Delbaen & Shige Peng & Emanuela Rosazza Gianin, 2010. "Representation of the penalty term of dynamic concave utilities," Finance and Stochastics, Springer, vol. 14(3), pages 449-472, September.
    2. 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.
    3. 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.
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    Cited by:

    1. Briand, Philippe & Elie, Romuald, 2013. "A simple constructive approach to quadratic BSDEs with or without delay," Stochastic Processes and their Applications, Elsevier, vol. 123(8), pages 2921-2939.
    2. Gregor Heyne & Michael Kupper & Christoph Mainberger, 2011. "Minimal Supersolutions of BSDEs with Lower Semicontinuous Generators," SFB 649 Discussion Papers SFB649DP2011-067, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
    3. Antonis Papapantoleon & Dylan Possamai & Alexandros Saplaouras, 2016. "Existence and uniqueness results for BSDEs with jumps: the whole nine yards," Papers 1607.04214, arXiv.org, revised Apr 2017.

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