RBSDEs with optional barriers: monotone approximation

Siham Bouhadou; Astrid Hilbert; Youssef Ouknine

doi:10.3934/puqr.2022005

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June 2022, 7(2): 67-84. doi: 10.3934/puqr.2022005

RBSDEs with optional barriers: monotone approximation

Siham Bouhadou ^1,, , Astrid Hilbert ^2, and Youssef Ouknine ^1,3,

1.	Department of Mathematics, Faculty of Sciences Semlalia, Cadi Ayyad University, Bd. Prince My Abdellah, B.P. 2390, 40000 Marrakech, Morocco
2.	Department of Mathematics, Linnaeus University, Vaxjo 351 95, Sweden
3.	Africa Business School, Mohammed VI Polytechnic University, Lot 660, Hay Moulay Rachid Ben Guerir, 43150, Morocco

^*Corresponding author

Received July 2021 Accepted April 03, 2022 Published June 2022 Early access May 2022

In this short note we consider reflected backward stochastic differential equations (RBSDEs) with a Lipschitz driver and barrier processes that are optional and right lower semicontinuous. In this case, the barrier is represented as a nondecreasing limit of right continuous with left limit (RCLL) barriers. We combine some well-known existence results for RCLL barriers with comparison arguments for the control process to construct solutions. Finally, we highlight the connection of these RBSDEs with standard RCLL BSDEs.

Keywords: Reflected backward stochastic differential equation, g-expectation, Optional barrier, Monotone approximation, Comparison principle.

Mathematics Subject Classification: 60H10, 60G40.

Citation: Siham Bouhadou, Astrid Hilbert, Youssef Ouknine. RBSDEs with optional barriers: monotone approximation. Probability, Uncertainty and Quantitative Risk, 2022, 7 (2) : 67-84. doi: 10.3934/puqr.2022005

References:

[1]	Barrieu, P. and El Karoui, N., Optimal derivatives design under dynamic risk measures, In: Math. Finance, Contemp. Math, Amer. Math. Soc., Providence, RI, 2004, 351: 13–25.
[2]	Bouhadou, S. and Ouknine, Y., Reflected BSDEs when the obstacle is predictable and nonlinear optimal stopping problem, Stochastics and Dynamics, 2021, 21(8): 2150049. doi: 10.1142/S0219493721500490.
[3]	Crépey, S. and Matoussi, A., Reflected and doubly reflected BSDEs with jumps: A priori estimates and comparison, The Annals of Applied Probability, 2008, 18(5): 2041−2069.
[4]	Delbaen, F., Peng, S. and Rosazza Gianin, E., Representation of the penalty term of dynamic concave utilities, Finance Stoch., 2010, 14(3): 449−472. doi: 10.1007/s00780-009-0119-7.
[5]	Dellacherie, C. and Meyer, P.-A., Probabilités et Potentiel, Théorie des Martingales, Chap. V-VIII, Hermann, 1980.
[6]	Dellacherie, C. and Lenglart, E., Sur les problèmes de régularisation, de recollement et d’interpolation en théorie des processus, Séminaire de Probabilités, 1982, 16: 298−313.
[7]	El Karoui, N., Les aspects probabilistes du contrôle stochastique, In: Hennequin, P. L. (ed.), Ecole d’Eté de Probabilités de Saint-Flour IX-1979, Lect. Notes in Math., 1981, 876: 73–238.
[8]	El Karoui, N., Kapoudjian, C., Pardoux, E., Peng, S. and Quenez, M.-C., Reflected solutions of Backward SDE’s and related obstacle problems for PDE’s, The Annals of Probability, 1997, 25(2): 702−737.
[9]	Essaky, H., Reflected backward stochastic differential equation with jumps and RCLL obstacle, Bulletin des Sciences Mathématiques, 2008, 132(8): 690−710.
[10]	Grigorova, M., Imkeller, P., Offen, E., Ouknine, Y. and Quenez, M.-C., Reflected BSDEs when the obstacle is not right-continuous and optimal stopping, The Annals of Applied Probability, 2017, 27(5): 3153−3188.
[11]	Grigorova, M., Imkeller, P., Ouknine, Y. and Quenez, M.-C., Optimal stopping with f-expectations: The irregular case, Stochastic Processes and their Applications, 2020, 130(3): 1258−1288. doi: 10.1016/j.spa.2019.05.001.
[12]	Hamadène, S., Reflected BSDE’s with discontinuous barrier and application, Stochastics and Stochastic Reports, 2002, 74(3/4): 571−596.
[13]	Hamadène, S. and Ouknine, Y., Reflected backward stochastic differential equation with jumps and random obstacle, Electronic Journal of Probability, 2003, 8: 1−20.
[14]	Hamadène, S. and Ouknine, Y., Reflected backward SDEs with general jumps, Teor. Veroyatnost. i Primenen., 2015, 60(2): 357−376. doi: 10.4213/tvp4623.
[15]	Hamadène, S. and Wang, H., BSDEs with two RCLL reflecting obstacles driven by a Brownian motion and Poisson measure and related mixed zero-sum games, Stochastic Processes and their Applications, 2009, 119(9): 2881−2912. doi: 10.1016/j.spa.2009.03.004.
[16]	Klimsiak, T., Rzymowski, M. and Słomiński, L., Reflected BSDEs with regulated trajectories, Stochastic Processes and their Applications, 2019, 129(4): 1153−1184. doi: 10.1016/j.spa.2018.04.011.
[17]	Kobylanski, M. and Quenez, M.-C., Optimal stopping time problem in a general framework, Electronic Journal of Probability, 2012, 17: 1−28.
[18]	Kobylanski, M., Quenez, M.-C. and De Campagnolle, M.-R., Dynkin games in a general framework, Stochastics, 2014, 86(2): 304−329. doi: 10.1080/17442508.2013.778860.
[19]	Neveu, J., Martingales à Temps Discret, Masson, Paris, 1972.
[20]	Peng, S., Nonlinear Expectations, Nonlinear Evaluations and Risk Measures, In: Frittelli, M. and Runggaldier, W. (eds.), Stochastic Methods in Finance, Lecture Notes in Math., Springer, Berlin, 2004, 1856: 165–253.
[21]	Peng, S. and Xu, M., The smallest g-supermartingale and reflected BSDE with single and double L² obstacle, Ann. I. H. Poincare, 2005, 41(3): 605−630.
[22]	Quenez, M.-C. and Sulem, A., BSDEs with jumps, optimization and applications to dynamic risk measures, Stoch. Proc. Appl., 2013, 123(8): 3328−3357. doi: 10.1016/j.spa.2013.02.016.
[23]	Quenez, M.-C. and Sulem, A., Reflected BSDEs and robust optimal stopping for dynamic risk measures with jumps, Stoch. Proc. Appl., 2014, 124(9): 3031−3054. doi: 10.1016/j.spa.2014.04.007.

show all references

References:

[1]	Barrieu, P. and El Karoui, N., Optimal derivatives design under dynamic risk measures, In: Math. Finance, Contemp. Math, Amer. Math. Soc., Providence, RI, 2004, 351: 13–25.
[2]	Bouhadou, S. and Ouknine, Y., Reflected BSDEs when the obstacle is predictable and nonlinear optimal stopping problem, Stochastics and Dynamics, 2021, 21(8): 2150049. doi: 10.1142/S0219493721500490.
[3]	Crépey, S. and Matoussi, A., Reflected and doubly reflected BSDEs with jumps: A priori estimates and comparison, The Annals of Applied Probability, 2008, 18(5): 2041−2069.
[4]	Delbaen, F., Peng, S. and Rosazza Gianin, E., Representation of the penalty term of dynamic concave utilities, Finance Stoch., 2010, 14(3): 449−472. doi: 10.1007/s00780-009-0119-7.
[5]	Dellacherie, C. and Meyer, P.-A., Probabilités et Potentiel, Théorie des Martingales, Chap. V-VIII, Hermann, 1980.
[6]	Dellacherie, C. and Lenglart, E., Sur les problèmes de régularisation, de recollement et d’interpolation en théorie des processus, Séminaire de Probabilités, 1982, 16: 298−313.
[7]	El Karoui, N., Les aspects probabilistes du contrôle stochastique, In: Hennequin, P. L. (ed.), Ecole d’Eté de Probabilités de Saint-Flour IX-1979, Lect. Notes in Math., 1981, 876: 73–238.
[8]	El Karoui, N., Kapoudjian, C., Pardoux, E., Peng, S. and Quenez, M.-C., Reflected solutions of Backward SDE’s and related obstacle problems for PDE’s, The Annals of Probability, 1997, 25(2): 702−737.
[9]	Essaky, H., Reflected backward stochastic differential equation with jumps and RCLL obstacle, Bulletin des Sciences Mathématiques, 2008, 132(8): 690−710.
[10]	Grigorova, M., Imkeller, P., Offen, E., Ouknine, Y. and Quenez, M.-C., Reflected BSDEs when the obstacle is not right-continuous and optimal stopping, The Annals of Applied Probability, 2017, 27(5): 3153−3188.
[11]	Grigorova, M., Imkeller, P., Ouknine, Y. and Quenez, M.-C., Optimal stopping with f-expectations: The irregular case, Stochastic Processes and their Applications, 2020, 130(3): 1258−1288. doi: 10.1016/j.spa.2019.05.001.
[12]	Hamadène, S., Reflected BSDE’s with discontinuous barrier and application, Stochastics and Stochastic Reports, 2002, 74(3/4): 571−596.
[13]	Hamadène, S. and Ouknine, Y., Reflected backward stochastic differential equation with jumps and random obstacle, Electronic Journal of Probability, 2003, 8: 1−20.
[14]	Hamadène, S. and Ouknine, Y., Reflected backward SDEs with general jumps, Teor. Veroyatnost. i Primenen., 2015, 60(2): 357−376. doi: 10.4213/tvp4623.
[15]	Hamadène, S. and Wang, H., BSDEs with two RCLL reflecting obstacles driven by a Brownian motion and Poisson measure and related mixed zero-sum games, Stochastic Processes and their Applications, 2009, 119(9): 2881−2912. doi: 10.1016/j.spa.2009.03.004.
[16]	Klimsiak, T., Rzymowski, M. and Słomiński, L., Reflected BSDEs with regulated trajectories, Stochastic Processes and their Applications, 2019, 129(4): 1153−1184. doi: 10.1016/j.spa.2018.04.011.
[17]	Kobylanski, M. and Quenez, M.-C., Optimal stopping time problem in a general framework, Electronic Journal of Probability, 2012, 17: 1−28.
[18]	Kobylanski, M., Quenez, M.-C. and De Campagnolle, M.-R., Dynkin games in a general framework, Stochastics, 2014, 86(2): 304−329. doi: 10.1080/17442508.2013.778860.
[19]	Neveu, J., Martingales à Temps Discret, Masson, Paris, 1972.
[20]	Peng, S., Nonlinear Expectations, Nonlinear Evaluations and Risk Measures, In: Frittelli, M. and Runggaldier, W. (eds.), Stochastic Methods in Finance, Lecture Notes in Math., Springer, Berlin, 2004, 1856: 165–253.
[21]	Peng, S. and Xu, M., The smallest g-supermartingale and reflected BSDE with single and double L² obstacle, Ann. I. H. Poincare, 2005, 41(3): 605−630.
[22]	Quenez, M.-C. and Sulem, A., BSDEs with jumps, optimization and applications to dynamic risk measures, Stoch. Proc. Appl., 2013, 123(8): 3328−3357. doi: 10.1016/j.spa.2013.02.016.
[23]	Quenez, M.-C. and Sulem, A., Reflected BSDEs and robust optimal stopping for dynamic risk measures with jumps, Stoch. Proc. Appl., 2014, 124(9): 3031−3054. doi: 10.1016/j.spa.2014.04.007.

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RBSDEs with optional barriers: monotone approximation

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