
Previous Article
Threshold reweighted Nadaraya–Watson estimation of jumpdiffusion models
 PUQR Home
 This Issue

Next Article
On the laws of the iterated logarithm with meanuncertainty under sublinear expectations
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. 
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 discretetime numerical scheme for quadratic reflected BSDEs and obtain the explicit rate of convergence by applying the quantitative stability result.
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., Discretetime 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., Discretetime approximation and MonteCarlo 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/s0044000604970. 
[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 regressionbased 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/s1140102102567. 
[19] 
Zhang, J., A numerical scheme for BSDEs, The Annals of Applied Probability, 2004, 14(1): 459−488. 
show all references
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., Discretetime 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., Discretetime approximation and MonteCarlo 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/s0044000604970. 
[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 regressionbased 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/s1140102102567. 
[19] 
Zhang, J., A numerical scheme for BSDEs, The Annals of Applied Probability, 2004, 14(1): 459−488. 
[1] 
Jiequn Han, Jihao Long. Convergence of the deep BSDE method for coupled FBSDEs. Probability, Uncertainty and Quantitative Risk, 2020, 5 (0) : 5. doi: 10.1186/s4154602000047w 
[2] 
Yifan Jiang, Jinfeng Li. Convergence of the Deep BSDE method for FBSDEs with nonLipschitz coefficients. Probability, Uncertainty and Quantitative Risk, 2021, 6 (4) : 391408. doi: 10.3934/puqr.2021019 
[3] 
Hélène Hibon, Ying Hu, Yiqing Lin, Peng Luo, Falei Wang. Quadratic BSDEs with mean reflection. Mathematical Control and Related Fields, 2018, 8 (3&4) : 721738. doi: 10.3934/mcrf.2018031 
[4] 
Yu A. Kutoyants. On approximation of BSDE and multistep MLEprocesses. Probability, Uncertainty and Quantitative Risk, 2016, 1 (0) : 4. doi: 10.1186/s4154601600050 
[5] 
Dmytro Marushkevych, Alexandre Popier. Limit behaviour of the minimal solution of a BSDE with singular terminal condition in the non Markovian setting. Probability, Uncertainty and Quantitative Risk, 2020, 5 (0) : 1. doi: 10.1186/s4154602000435 
[6] 
Zigen Ouyang, Chunhua Ou. Global stability and convergence rate of traveling waves for a nonlocal model in periodic media. Discrete and Continuous Dynamical Systems  B, 2012, 17 (3) : 9931007. doi: 10.3934/dcdsb.2012.17.993 
[7] 
Oleg Makarenkov, Paolo Nistri. On the rate of convergence of periodic solutions in perturbed autonomous systems as the perturbation vanishes. Communications on Pure and Applied Analysis, 2008, 7 (1) : 4961. doi: 10.3934/cpaa.2008.7.49 
[8] 
Zhilei Liang. Convergence rate of solutions to the contact discontinuity for the compressible NavierStokes equations. Communications on Pure and Applied Analysis, 2013, 12 (5) : 19071926. doi: 10.3934/cpaa.2013.12.1907 
[9] 
Yulan Lu, Minghui Song, Mingzhu Liu. Convergence rate and stability of the splitstep theta method for stochastic differential equations with piecewise continuous arguments. Discrete and Continuous Dynamical Systems  B, 2019, 24 (2) : 695717. doi: 10.3934/dcdsb.2018203 
[10] 
Haibo Cui, Haiyan Yin. Convergence rate of solutions toward stationary solutions to the isentropic micropolar fluid model in a half line. Discrete and Continuous Dynamical Systems  B, 2021, 26 (6) : 28992920. doi: 10.3934/dcdsb.2020210 
[11] 
Tohru Nakamura, Shinya Nishibata, Naoto Usami. Convergence rate of solutions towards the stationary solutions to symmetric hyperbolicparabolic systems in half space. Kinetic and Related Models, 2018, 11 (4) : 757793. doi: 10.3934/krm.2018031 
[12] 
Tong Li, Hui Yin. Convergence rate to strong boundary layer solutions for generalized BBMBurgers equations with nonconvex flux. Communications on Pure and Applied Analysis, 2014, 13 (2) : 835858. doi: 10.3934/cpaa.2014.13.835 
[13] 
Hanchun Yang, Meimei Zhang, Qin Wang. Global solutions of shock reflection problem for the pressure gradient system. Communications on Pure and Applied Analysis, 2020, 19 (6) : 33873428. doi: 10.3934/cpaa.2020150 
[14] 
Haibo Cui, Zhensheng Gao, Haiyan Yin, Peixing Zhang. Stationary waves to the twofluid nonisentropic NavierStokesPoisson system in a half line: Existence, stability and convergence rate. Discrete and Continuous Dynamical Systems, 2016, 36 (9) : 48394870. doi: 10.3934/dcds.2016009 
[15] 
Haiyan Yin, Changjiang Zhu. Convergence rate of solutions toward stationary solutions to a viscous liquidgas twophase flow model in a half line. Communications on Pure and Applied Analysis, 2015, 14 (5) : 20212042. doi: 10.3934/cpaa.2015.14.2021 
[16] 
Luis D'Alto, Martin Corless. Incremental quadratic stability. Numerical Algebra, Control and Optimization, 2013, 3 (1) : 175201. doi: 10.3934/naco.2013.3.175 
[17] 
Jinyan Fan, Jianyu Pan. On the convergence rate of the inexact LevenbergMarquardt method. Journal of Industrial and Management Optimization, 2011, 7 (1) : 199210. doi: 10.3934/jimo.2011.7.199 
[18] 
Shahad Alazzawi, Jicheng Liu, Xianming Liu. Convergence rate of synchronization of systems with additive noise. Discrete and Continuous Dynamical Systems  B, 2017, 22 (2) : 227245. doi: 10.3934/dcdsb.2017012 
[19] 
Armand Bernou. A semigroup approach to the convergence rate of a collisionless gas. Kinetic and Related Models, 2020, 13 (6) : 10711106. doi: 10.3934/krm.2020038 
[20] 
Yves Bourgault, Damien Broizat, PierreEmmanuel Jabin. Convergence rate for the method of moments with linear closure relations. Kinetic and Related Models, 2015, 8 (1) : 127. doi: 10.3934/krm.2015.8.1 
Impact Factor:
Tools
Metrics
Other articles
by authors
[Back to Top]