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Quadratic Reflected BSDEs with Unbounded Obstacles

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  • Erhan Bayraktar
  • Song Yao

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, comparison, 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 convex or concave in the $z$-variable.

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  • Erhan Bayraktar & Song Yao, 2010. "Quadratic Reflected BSDEs with Unbounded Obstacles," Papers 1005.3565, arXiv.org, revised Mar 2011.
    • Handle: RePEc:arx:papers:1005.3565
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    References listed on IDEAS

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    1. 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.
    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. Bayraktar, Erhan & Yao, Song, 2011. "Optimal stopping for non-linear expectations--Part II," Stochastic Processes and their Applications, Elsevier, vol. 121(2), pages 212-264, February.
    4. Erhan Bayraktar & Song Yao, 2009. "Optimal Stopping for Non-linear Expectations," Papers 0905.3601, arXiv.org, revised Jan 2011.
    5. 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.
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    1. 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.
    2. Keller, Christian, 2016. "Viscosity solutions of path-dependent integro-differential equations," Stochastic Processes and their Applications, Elsevier, vol. 126(9), pages 2665-2718.
    3. Dingqian Sun, 2020. "The convergence rate from discrete to continuous optimal investment stopping problem," Papers 2004.14627, arXiv.org.
    4. 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.
    5. 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.
    6. Edward Kim & Tianyang Nie & Marek Rutkowski, 2018. "Arbitrage-free pricing of American options in nonlinear markets," Papers 1804.10753, arXiv.org, revised Jul 2018.

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