# American Institute of Mathematical Sciences

September 2018, 8(3&4): 721-738. doi: 10.3934/mcrf.2018031

## Quadratic BSDEs with mean reflection

 1 Institut de Recherche Mathématique de Rennes, Université Rennes 1, 35042 Rennes Cedex, France 2 School of Mathematical Sciences, Fudan University, Shanghai 200433, China 3 School of Mathematical Sciences, Shanghai Jiao Tong University, 200240 Shanghai, China 4 Centre de Mathématiques Appliquées, École Polytechnique, 91128 Palaiseau Cedex, France 5 Department of Mathematics, ETH Zurich, 8092 Zurich, Switzerland 6 Zhongtai Securities Institute for Financial Studies and Institute for Advanced Research, Shandong University, Jinan 250100, China

* Corresponding authorr: Y. Hu

Received  May 2017 Revised  October 2017 Published  September 2018

Fund Project: Y. Hu's research is partially supported by Lebesgue Center of Mathematics "Investissements d'avenir" Program (No. ANR-11-LABX-0020-01), by ANR CAESARS (No. ANR-15-CE05-0024) and by ANR MFG (No. ANR-16-CE40-0015-01). Y. Lin's research is partially supported by the European Research Council under grant 321111. P. Luo's research is partially supported by the National Science Foundation of China "Research Fund for International Young Scientists"(No. 11550110184) and by National Natural Science Foundation of China (No. 11671257). F. Wang's research is partially supported by the National Natural Science Foundation of China (No.11601282 and 11526205), by the Shandong Province Natural Science Foundation (No. ZR2016AQ10) and by the China Scholarship Council (No. 201606225002)

The present paper is devoted to the study of the well-posedness of BSDEs with mean reflection whenever the generator has quadratic growth in the $z$ argument. This work is the sequel of [6] in which a notion of BSDEs with mean reflection is developed to tackle the super-hedging problem under running risk management constraints. By the contraction mapping argument, we first prove that the quadratic BSDE with mean reflection admits a unique deterministic flat local solution on a small time interval whenever the terminal value is bounded. Moreover, we build the global solution on the whole time interval by stitching local solutions when the generator is uniformly bounded with respect to the $y$ argument.

Citation: Hélène Hibon, Ying Hu, Yiqing Lin, Peng Luo, Falei Wang. Quadratic BSDEs with mean reflection. Mathematical Control & Related Fields, 2018, 8 (3&4) : 721-738. doi: 10.3934/mcrf.2018031
##### References:
 [1] S. Ankirchner, P. Imkeller and G. dos Reis, Classical and variational differentiability of BSDEs with quadratic growth, Electron. J. Probab, 12 (2007), 1418-1453. doi: 10.1214/EJP.v12-462. [2] P. Barrieu and N. El Karoui, Monotone stability of quadratic semimartingales with applications to unbounded general quadratic BSDEs, Ann. Probab., 41 (2013), 1831-1863. doi: 10.1214/12-AOP743. [3] B. Bouchard, R. Elie and A. Réveillac, BSDEs with weak terminal condition, Ann. Probab., 43 (2015), 572-604. doi: 10.1214/14-AOP913. [4] P. Briand, P. E. Chaudru de Raynal, A. Guillin and C. Labart, Particles systems and numerical schemes for mean reflected stochastic differential equations, preprint, arXiv: 1612.06886 [5] P. Briand and F. Confortola, BSDEs with stochastic Lipschitz condition and quadratic PDEs in Hilbert spaces, Stochastic Process. Appl., 118 (2008), 818-838. doi: 10.1016/j.spa.2007.06.006. [6] P. Briand, R. Elie and Y. Hu, BSDEs with mean reflection, Ann. Appl. Probab., 28 (2018), 482–510, arXiv: 1605.06301 doi: 10.1214/17-AAP1310. [7] P. Briand and R. Elie, A simple constructive approach to quadratic BSDEs with or without delay, Stochastic Process. Appl., 123 (2013), 2921-2939. doi: 10.1016/j.spa.2013.02.013. [8] P. Briand and Y. Hu, BSDE with quadratic growth and unbounded terminal value, Probab. Theory Related Fields, 136 (2006), 604-618. doi: 10.1007/s00440-006-0497-0. [9] P. Briand and Y. Hu, Quadratic BSDEs with convex generators and unbounded terminal conditions, Probab. Theory Related Fields, 141 (2008), 543-567. doi: 10.1007/s00440-007-0093-y. [10] R. Buckdahn and Y. Hu, Pricing of American contingent claims with jump stock price and constrained portfolios, Math. Oper. Res., 23 (1998), 177-203. doi: 10.1287/moor.23.1.177. [11] R. Buckdahn and Y. Hu, Hedging contingent claims for a large investor in an incomplete market, Adv. in Appl. Probab., 30 (1998), 239-255. doi: 10.1239/aap/1035228002. [12] J. F. Chassagneux, R. Elie and I. Kharroubi, A note on existence and uniqueness for solutions of multidimensional reflected BSDEs, Electron. Commun. Probab., 16 (2011), 120-128. doi: 10.1214/ECP.v16-1614. [13] P. Cheridito and K. Nam, BSDEs with terminal conditions that have bounded Malliavin derivative, J. Funct. Anal., 266 (2014), 1257-1285. doi: 10.1016/j.jfa.2013.12.004. [14] P. Cheridito and K. Nam, Multidimensional quadratic and subquadratic BSDEs with special structure, Stochastics, 87 (2015), 871-884. doi: 10.1080/17442508.2015.1013959. [15] J. Cvitanić and I. Karatzas, Backward stochastic differential equations with reflection and Dynkin games, Ann. Probab., 24 (1996), 2024-2056. doi: 10.1214/aop/1041903216. [16] J. Cvitanić, I. Karatzas and H. M. Soner, Backward stochastic differential equations with constraints on the gains-process, Ann. Probab., 26 (1998), 1522-1551. doi: 10.1214/aop/1022855872. [17] N. El Karoui, C. Kapoudjian, E. Pardoux, S. Peng and M. C. Quenez, Reflected solutions of backward SDE's, and related obstacle problems for PDE's, Ann. Probab., 25 (1997), 702-737. doi: 10.1214/aop/1024404416. [18] N. El Karoui, E. Pardoux and M. C. Quenez, Reflected backward SDEs and American options, in Numerical Methods in Finance (eds. L. C. G. Rogers and D. Talay), Cambridge Univ. Press, 13 (1997), 215–231. [19] N. El Karoui, S. Peng and M. C. Quenez, Backward stochastic differential equations in finance, Math. Finance, 7 (1997), 1-71. doi: 10.1111/1467-9965.00022. [20] C. Frei and G. dos Reis, A financial market with interatcting investors: Does an equilibrium exist?, Math. Financ. Econ., 4 (2011), 161-182. doi: 10.1007/s11579-011-0039-0. [21] S. Hamadene and M. Jeanblanc, On the starting and stopping problem: Application in reversible investments, Math. Oper. Res., 32 (2007), 182-192. doi: 10.1287/moor.1060.0228. [22] S. Hamadene and J. Zhang, Switching problem and related system of reflected backward SDEs, Stochastic Process. Appl., 120 (2010), 403-426. doi: 10.1016/j.spa.2010.01.003. [23] J. Harter and A. Richou, A stability approach for solving multidimensional quadratic BSDEs, preprint, arXiv: 1606.08627 [24] Y. Hu, P. Imkeller and M. Müller, Utility maximization in incomplete markets, Ann. Appl. Probab., 15 (2005), 1691-1712. doi: 10.1214/105051605000000188. [25] Y. Hu and S. Tang, Multi-dimensional BSDE with oblique reflection and optimal switching, Probab. Theory Related Fields, 147 (2010), 89-121. doi: 10.1007/s00440-009-0202-1. [26] Y. Hu and S. Tang, Multi-dimensional backward stochastic differential equations of diagonally quadratic generators, Stochastic Process. Appl., 126 (2016), 1066-1086. doi: 10.1016/j.spa.2015.10.011. [27] N. Kazamaki, Continuous Exponential Martingales and BMO, Springer-Verlag, Berlin, 1994. doi: 10.1007/BFb0073585. [28] M. Kobylanski, Backward stochastic differential equations and partial differential equations with quadratic growth, Ann. Probab., 28 (2000), 558-602. doi: 10.1214/aop/1019160253. [29] M.-A. Morlais, Quadratic BSDEs driven by a continuous martingale and applications to the utility maximization problem, Finance Stoch., 13 (2009), 121-150. doi: 10.1007/s00780-008-0079-3. [30] D. Nualart, The Malliavin Calculus and Related Topics, 2$^{nd}$ edition, Springer, Berlin, 2006. doi: 10.1007/3-540-28329-3. [31] E. Pardoux and S. Peng, Adapted solution of a backward stochastic differential equation, Systems Control Lett., 14 (1990), 55-61. doi: 10.1016/0167-6911(90)90082-6. [32] S. Peng and M. Xu, Reflected BSDE with a constraint and its applications in an incomplete market, Bernoulli, 16 (2010), 614-640. doi: 10.3150/09-BEJ227. [33] R. Tevzadze, Solvability of backward stochastic differential equations with quadratic growth, Stochastic Process. Appl., 118 (2008), 503-515. doi: 10.1016/j.spa.2007.05.009. [34] H. Xing and G. Zitkovic, A class of globally solvable Markovian quadratic BSDE systems and applications, Ann. Probab., 46 (2018), 491–550, arXiv: 1603.00217 doi: 10.1214/17-AOP1190.

show all references

##### References:
 [1] S. Ankirchner, P. Imkeller and G. dos Reis, Classical and variational differentiability of BSDEs with quadratic growth, Electron. J. Probab, 12 (2007), 1418-1453. doi: 10.1214/EJP.v12-462. [2] P. Barrieu and N. El Karoui, Monotone stability of quadratic semimartingales with applications to unbounded general quadratic BSDEs, Ann. Probab., 41 (2013), 1831-1863. doi: 10.1214/12-AOP743. [3] B. Bouchard, R. Elie and A. Réveillac, BSDEs with weak terminal condition, Ann. Probab., 43 (2015), 572-604. doi: 10.1214/14-AOP913. [4] P. Briand, P. E. Chaudru de Raynal, A. Guillin and C. Labart, Particles systems and numerical schemes for mean reflected stochastic differential equations, preprint, arXiv: 1612.06886 [5] P. Briand and F. Confortola, BSDEs with stochastic Lipschitz condition and quadratic PDEs in Hilbert spaces, Stochastic Process. Appl., 118 (2008), 818-838. doi: 10.1016/j.spa.2007.06.006. [6] P. Briand, R. Elie and Y. Hu, BSDEs with mean reflection, Ann. Appl. Probab., 28 (2018), 482–510, arXiv: 1605.06301 doi: 10.1214/17-AAP1310. [7] P. Briand and R. Elie, A simple constructive approach to quadratic BSDEs with or without delay, Stochastic Process. Appl., 123 (2013), 2921-2939. doi: 10.1016/j.spa.2013.02.013. [8] P. Briand and Y. Hu, BSDE with quadratic growth and unbounded terminal value, Probab. Theory Related Fields, 136 (2006), 604-618. doi: 10.1007/s00440-006-0497-0. [9] P. Briand and Y. Hu, Quadratic BSDEs with convex generators and unbounded terminal conditions, Probab. Theory Related Fields, 141 (2008), 543-567. doi: 10.1007/s00440-007-0093-y. [10] R. Buckdahn and Y. Hu, Pricing of American contingent claims with jump stock price and constrained portfolios, Math. Oper. Res., 23 (1998), 177-203. doi: 10.1287/moor.23.1.177. [11] R. Buckdahn and Y. Hu, Hedging contingent claims for a large investor in an incomplete market, Adv. in Appl. Probab., 30 (1998), 239-255. doi: 10.1239/aap/1035228002. [12] J. F. Chassagneux, R. Elie and I. Kharroubi, A note on existence and uniqueness for solutions of multidimensional reflected BSDEs, Electron. Commun. Probab., 16 (2011), 120-128. doi: 10.1214/ECP.v16-1614. [13] P. Cheridito and K. Nam, BSDEs with terminal conditions that have bounded Malliavin derivative, J. Funct. Anal., 266 (2014), 1257-1285. doi: 10.1016/j.jfa.2013.12.004. [14] P. Cheridito and K. Nam, Multidimensional quadratic and subquadratic BSDEs with special structure, Stochastics, 87 (2015), 871-884. doi: 10.1080/17442508.2015.1013959. [15] J. Cvitanić and I. Karatzas, Backward stochastic differential equations with reflection and Dynkin games, Ann. Probab., 24 (1996), 2024-2056. doi: 10.1214/aop/1041903216. [16] J. Cvitanić, I. Karatzas and H. M. Soner, Backward stochastic differential equations with constraints on the gains-process, Ann. Probab., 26 (1998), 1522-1551. doi: 10.1214/aop/1022855872. [17] N. El Karoui, C. Kapoudjian, E. Pardoux, S. Peng and M. C. Quenez, Reflected solutions of backward SDE's, and related obstacle problems for PDE's, Ann. Probab., 25 (1997), 702-737. doi: 10.1214/aop/1024404416. [18] N. El Karoui, E. Pardoux and M. C. Quenez, Reflected backward SDEs and American options, in Numerical Methods in Finance (eds. L. C. G. Rogers and D. Talay), Cambridge Univ. Press, 13 (1997), 215–231. [19] N. El Karoui, S. Peng and M. C. Quenez, Backward stochastic differential equations in finance, Math. Finance, 7 (1997), 1-71. doi: 10.1111/1467-9965.00022. [20] C. Frei and G. dos Reis, A financial market with interatcting investors: Does an equilibrium exist?, Math. Financ. Econ., 4 (2011), 161-182. doi: 10.1007/s11579-011-0039-0. [21] S. Hamadene and M. Jeanblanc, On the starting and stopping problem: Application in reversible investments, Math. Oper. Res., 32 (2007), 182-192. doi: 10.1287/moor.1060.0228. [22] S. Hamadene and J. Zhang, Switching problem and related system of reflected backward SDEs, Stochastic Process. Appl., 120 (2010), 403-426. doi: 10.1016/j.spa.2010.01.003. [23] J. Harter and A. Richou, A stability approach for solving multidimensional quadratic BSDEs, preprint, arXiv: 1606.08627 [24] Y. Hu, P. Imkeller and M. Müller, Utility maximization in incomplete markets, Ann. Appl. Probab., 15 (2005), 1691-1712. doi: 10.1214/105051605000000188. [25] Y. Hu and S. Tang, Multi-dimensional BSDE with oblique reflection and optimal switching, Probab. Theory Related Fields, 147 (2010), 89-121. doi: 10.1007/s00440-009-0202-1. [26] Y. Hu and S. Tang, Multi-dimensional backward stochastic differential equations of diagonally quadratic generators, Stochastic Process. Appl., 126 (2016), 1066-1086. doi: 10.1016/j.spa.2015.10.011. [27] N. Kazamaki, Continuous Exponential Martingales and BMO, Springer-Verlag, Berlin, 1994. doi: 10.1007/BFb0073585. [28] M. Kobylanski, Backward stochastic differential equations and partial differential equations with quadratic growth, Ann. Probab., 28 (2000), 558-602. doi: 10.1214/aop/1019160253. [29] M.-A. Morlais, Quadratic BSDEs driven by a continuous martingale and applications to the utility maximization problem, Finance Stoch., 13 (2009), 121-150. doi: 10.1007/s00780-008-0079-3. [30] D. Nualart, The Malliavin Calculus and Related Topics, 2$^{nd}$ edition, Springer, Berlin, 2006. doi: 10.1007/3-540-28329-3. [31] E. Pardoux and S. Peng, Adapted solution of a backward stochastic differential equation, Systems Control Lett., 14 (1990), 55-61. doi: 10.1016/0167-6911(90)90082-6. [32] S. Peng and M. Xu, Reflected BSDE with a constraint and its applications in an incomplete market, Bernoulli, 16 (2010), 614-640. doi: 10.3150/09-BEJ227. [33] R. Tevzadze, Solvability of backward stochastic differential equations with quadratic growth, Stochastic Process. Appl., 118 (2008), 503-515. doi: 10.1016/j.spa.2007.05.009. [34] H. Xing and G. Zitkovic, A class of globally solvable Markovian quadratic BSDE systems and applications, Ann. Probab., 46 (2018), 491–550, arXiv: 1603.00217 doi: 10.1214/17-AOP1190.
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