# American Institute of Mathematical Sciences

March  2022, 7(1): 13-30. doi: 10.3934/puqr.2022002

## Quantitative stability and numerical analysis of Markovian quadratic BSDEs with reflection

 1 Department of Finance and Control Sciences, School of Mathematical Sciences, Fudan University, Shanghai 200433, China 2 Department of Statistics, University of Warwick, Coventry, CV4 7AL, U.K.

g.liang@warwick.ac.uk

Received  December 06, 2021 Accepted  March 07, 2022 Published  March 2022 Early access  March 2022

Fund Project: The authors thank the editor and both referees for their careful reading and helpful comments. Dingqian Sun is partially supported by China Scholarship Council. Gechun Liang is partially supported by the National Natural Science Foundation of China (Grant No. 12171169) and Guangdong Basic and Applied Basic Research Foundation(Grant No. 2019A1515011338). GL thanks J. F. Chassagneux and A. Richou for helpful and inspiring discussions on how to extend to the state dependent volatility case. Shanjian Tang is partially supported by National Science Foundation of China (Grant No. 11631004) and National Key R&D Program of China (Grant No. 2018YFA0703903).

We study the quantitative stability of solutions to Markovian quadratic reflected backward stochastic differential equations (BSDEs) with bounded terminal data. By virtue of bounded mean oscillation martingale and change of measure techniques, we obtain stability estimates for the variation of the solutions with different underlying forward processes. In addition, we propose a truncated discrete-time numerical scheme for quadratic reflected BSDEs and obtain the explicit rate of convergence by applying the quantitative stability result.

Citation: Dingqian Sun, Gechun Liang, Shanjian Tang. Quantitative stability and numerical analysis of Markovian quadratic BSDEs with reflection. Probability, Uncertainty and Quantitative Risk, 2022, 7 (1) : 13-30. doi: 10.3934/puqr.2022002
##### References:
 [1] Bayraktar, E. and Yao, S., Quadratic reflected BSDEs with unbounded obstacles, Stochastic Processes and their Applications, 2012, 122(4): 1155−1203. doi: 10.1016/j.spa.2011.12.013. [2] Bouchard, B. and Chassagneux, J. F., Discrete-time approximation for continuously and discretely reflected BSDEs, Stochastic Processes and their Applications, 2008, 118(12): 2269−2293. doi: 10.1016/j.spa.2007.12.007. [3] Bouchard, B. and Touzi, N., Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations, Stochastic Processes and their Applications, 2004, 111(2): 175−206. doi: 10.1016/j.spa.2004.01.001. [4] Briand, P. and Hu, Y., BSDE with quadratic growth and unbounded terminal value, Probability Theory and Related Fields, 2006, 136(4): 604−618. doi: 10.1007/s00440-006-0497-0. [5] Briand, P. and Hu, Y., Quadratic BSDEs with convex generators and unbounded terminal conditions, Probability Theory and Related Fields, 2008, 141(3/4): 543−567. [6] Chassagneux, J. F. and Richou, A., Numerical simulation of quadratic BSDEs, The Annals of Applied Probability, 2016, 26(1): 262−304. [7] Gobet, E., Lemor, J. P. and Warin, X., A regression-based Monte Carlo method to solve backward stochastic differential equations, The Annals of Applied Probability, 2005, 15(3): 2172−2202. [8] Hu, Y., Li, X. and Wen, J., Anticipated backward stochastic differential equations with quadratic growth, Journal of Differential Equations, 2021, 270: 1298−1331. doi: 10.1016/j.jde.2020.07.001. [9] Imkeller, P. and Dos Reis, G., Path regularity and explicit convergence rate for BSDE with truncated quadratic growth, Stochastic Processes and their Applications, 2010, 120(3): 348−379. doi: 10.1016/j.spa.2009.11.004. [10] 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. [11] Lepeltier, J. P. and Xu, M., Reflected BSDE with quadratic growth and unbounded terminal value, arXiv: 0711.0619v1, 2007. [12] Kazamaki, N., Continuous Exponential Martingales and BMO, Lecture Notes in Mathematics, Springer, Berlin, Heidelberg, 1994. [13] Kloeden, P. E. and Platen, E., Numerical Solution of Stochastic Differential Equations, Applications of Mathematics, Springer, Berlin, 1992. [14] Kobylanski, M., Backward stochastic differential equations and partial differential equations with quadratic growth, The Annals of Probability, 2000, 28(2): 558−602. [15] Kobylanski, M., Lepeltier, J. P., Quenez, M. C. and Torres, S., Reflected BSDE with superlinear quadratic coefficient, Probability and Mathematical Statistics, 2002, 22(1): 51−83. [16] Ma, J. and Zhang, J., Representations and regularities for solutions to BSDEs with reflections, Stochastic Processes and their Applications, 2005, 115(4): 539−569. doi: 10.1016/j.spa.2004.05.010. [17] Richou, A., Numerical simulation of BSDEs with drivers of quadratic growth, The Annals of Applied Probability, 2011, 21(5): 1933−1964. [18] Sun, D. Q., The convergence rate from discrete to continuous optimal investment stopping problem, Chinese Annals of Mathematics, Series B, 2021, 42(2): 259−280. doi: 10.1007/s11401-021-0256-7. [19] Zhang, J., A numerical scheme for BSDEs, The Annals of Applied Probability, 2004, 14(1): 459−488.

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##### References:
 [1] Bayraktar, E. and Yao, S., Quadratic reflected BSDEs with unbounded obstacles, Stochastic Processes and their Applications, 2012, 122(4): 1155−1203. doi: 10.1016/j.spa.2011.12.013. [2] Bouchard, B. and Chassagneux, J. F., Discrete-time approximation for continuously and discretely reflected BSDEs, Stochastic Processes and their Applications, 2008, 118(12): 2269−2293. doi: 10.1016/j.spa.2007.12.007. [3] Bouchard, B. and Touzi, N., Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations, Stochastic Processes and their Applications, 2004, 111(2): 175−206. doi: 10.1016/j.spa.2004.01.001. [4] Briand, P. and Hu, Y., BSDE with quadratic growth and unbounded terminal value, Probability Theory and Related Fields, 2006, 136(4): 604−618. doi: 10.1007/s00440-006-0497-0. [5] Briand, P. and Hu, Y., Quadratic BSDEs with convex generators and unbounded terminal conditions, Probability Theory and Related Fields, 2008, 141(3/4): 543−567. [6] Chassagneux, J. F. and Richou, A., Numerical simulation of quadratic BSDEs, The Annals of Applied Probability, 2016, 26(1): 262−304. [7] Gobet, E., Lemor, J. P. and Warin, X., A regression-based Monte Carlo method to solve backward stochastic differential equations, The Annals of Applied Probability, 2005, 15(3): 2172−2202. [8] Hu, Y., Li, X. and Wen, J., Anticipated backward stochastic differential equations with quadratic growth, Journal of Differential Equations, 2021, 270: 1298−1331. doi: 10.1016/j.jde.2020.07.001. [9] Imkeller, P. and Dos Reis, G., Path regularity and explicit convergence rate for BSDE with truncated quadratic growth, Stochastic Processes and their Applications, 2010, 120(3): 348−379. doi: 10.1016/j.spa.2009.11.004. [10] 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. [11] Lepeltier, J. P. and Xu, M., Reflected BSDE with quadratic growth and unbounded terminal value, arXiv: 0711.0619v1, 2007. [12] Kazamaki, N., Continuous Exponential Martingales and BMO, Lecture Notes in Mathematics, Springer, Berlin, Heidelberg, 1994. [13] Kloeden, P. E. and Platen, E., Numerical Solution of Stochastic Differential Equations, Applications of Mathematics, Springer, Berlin, 1992. [14] Kobylanski, M., Backward stochastic differential equations and partial differential equations with quadratic growth, The Annals of Probability, 2000, 28(2): 558−602. [15] Kobylanski, M., Lepeltier, J. P., Quenez, M. C. and Torres, S., Reflected BSDE with superlinear quadratic coefficient, Probability and Mathematical Statistics, 2002, 22(1): 51−83. [16] Ma, J. and Zhang, J., Representations and regularities for solutions to BSDEs with reflections, Stochastic Processes and their Applications, 2005, 115(4): 539−569. doi: 10.1016/j.spa.2004.05.010. [17] Richou, A., Numerical simulation of BSDEs with drivers of quadratic growth, The Annals of Applied Probability, 2011, 21(5): 1933−1964. [18] Sun, D. Q., The convergence rate from discrete to continuous optimal investment stopping problem, Chinese Annals of Mathematics, Series B, 2021, 42(2): 259−280. doi: 10.1007/s11401-021-0256-7. [19] Zhang, J., A numerical scheme for BSDEs, The Annals of Applied Probability, 2004, 14(1): 459−488.
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