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Quadratic reflected BSDEs with unbounded obstacles

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  • Bayraktar, Erhan
  • Yao, Song

Abstract

In this paper, we analyze a real-valued reflected backward stochastic differential equation (RBSDE) with an unbounded obstacle and an unbounded terminal condition when its generator f has quadratic growth in the z-variable. In particular, we obtain existence, uniqueness, and stability results, and consider the optimal stopping for quadratic g-evaluations. As an application of our results we analyze the obstacle problem for semi-linear parabolic PDEs in which the non-linearity appears as the square of the gradient. Finally, we prove a comparison theorem for these obstacle problems when the generator is concave in the z-variable.

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  • Bayraktar, Erhan & Yao, Song, 2012. "Quadratic reflected BSDEs with unbounded obstacles," Stochastic Processes and their Applications, Elsevier, vol. 122(4), pages 1155-1203.
    • Handle: RePEc:eee:spapps:v:122:y:2012:i:4:p:1155-1203
    DOI: 10.1016/j.spa.2011.12.013
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    1. Bayraktar, Erhan & Yao, Song, 2011. "Optimal stopping for non-linear expectations--Part I," Stochastic Processes and their Applications, Elsevier, vol. 121(2), pages 185-211, February.
    2. Matoussi, Anis, 1997. "Reflected solutions of backward stochastic differential equations with continuous coefficient," Statistics & Probability Letters, Elsevier, vol. 34(4), pages 347-354, June.
    3. Lepeltier, J. P. & San Martin, J., 1997. "Backward stochastic differential equations with continuous coefficient," Statistics & Probability Letters, Elsevier, vol. 32(4), pages 425-430, April.
    4. 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.
    5. Erhan Bayraktar & Song Yao, 2009. "Optimal Stopping for Non-linear Expectations," Papers 0905.3601, arXiv.org, revised Jan 2011.
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    Cited by:

    1. Edward Kim & Tianyang Nie & Marek Rutkowski, 2018. "Arbitrage-free pricing of American options in nonlinear markets," Papers 1804.10753, arXiv.org, revised Jul 2018.
    2. Bayraktar, Erhan & Yao, Song, 2017. "Optimal stopping with random maturity under nonlinear expectations," Stochastic Processes and their Applications, Elsevier, vol. 127(8), pages 2586-2629.
    3. Keller, Christian, 2016. "Viscosity solutions of path-dependent integro-differential equations," Stochastic Processes and their Applications, Elsevier, vol. 126(9), pages 2665-2718.
    4. Lionnet, Arnaud, 2014. "Some results on general quadratic reflected BSDEs driven by a continuous martingale," Stochastic Processes and their Applications, Elsevier, vol. 124(3), pages 1275-1302.
    5. Bayraktar, Erhan & Yao, Song, 2015. "Doubly reflected BSDEs with integrable parameters and related Dynkin games," Stochastic Processes and their Applications, Elsevier, vol. 125(12), pages 4489-4542.
    6. Dingqian Sun, 2020. "The convergence rate from discrete to continuous optimal investment stopping problem," Papers 2004.14627, arXiv.org.

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