# American Institute of Mathematical Sciences

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

## RBSDEs with optional barriers: monotone approximation

 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.

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.

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##### 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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