# American Institute of Mathematical Sciences

September  2015, 5(3): 501-516. doi: 10.3934/mcrf.2015.5.501

## Controlled reflected mean-field backward stochastic differential equations coupled with value function and related PDEs

 1 School of Mathematics and Statistics, Shandong University, Weihai, Weihai, 264209, China, China

Received  February 2014 Revised  February 2015 Published  July 2015

In this paper, we consider a new type of reflected mean-field backward stochastic differential equations (reflected MFBSDEs, for short), namely, controlled reflected MFBSDEs involving their value function. The existence and the uniqueness of the solution of such equation are proved by using an approximation method. We also adapt this method to give a comparison theorem for our reflected MFBSDEs. The related dynamic programming principle is obtained by extending the approach of stochastic backward semigroups introduced by Peng [11] in 1997. Finally, we show that the value function which our reflected MFBSDE is coupled with is the unique viscosity solution of the related nonlocal parabolic partial differential equation with obstacle.
Citation: Juan Li, Wenqiang Li. Controlled reflected mean-field backward stochastic differential equations coupled with value function and related PDEs. Mathematical Control and Related Fields, 2015, 5 (3) : 501-516. doi: 10.3934/mcrf.2015.5.501
##### References:
 [1] G. Barles, R. Buckdahn and E. Pardoux, Backward stochastic differential equations and integral-partial differential equations, Stoch. Stoch. Rep., 60 (1997), 57-83. doi: 10.1080/17442509708834099. [2] R. Buckdahn and J. Li, Stochastic differential games and viscosity solutions of Hamilton- Jacobi-Bellman-Isaacs equations, SIAM. J. Control. Optim., 47 (2008), 444-475. doi: 10.1137/060671954. [3] R. Buckdahn and J. Li, Stochastic differential games with reflection and related obstacle problems for Isaacs equations, Acta Math. Appl. Sin.-Enql. Ser., 27 (2011), 647-678. doi: 10.1007/s10255-011-0068-8. [4] R. Buckdahn, B. Djehiche, J. Li and S. Peng, Mean-field backward stochastic differential equations. A limit approach, Ann. Probab., 37 (2009), 1524-1565. doi: 10.1214/08-AOP442. [5] R. Buckdahn, J. Li and S. Peng, Mean-field backward stochastic differential equations and related partial differential equations, Stoch. Proc. App., 119 (2009), 3133-3154. doi: 10.1016/j.spa.2009.05.002. [6] M. G. Crandall, H. Ishii and P. L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc., 27 (1992), 1-67. doi: 10.1090/S0273-0979-1992-00266-5. [7] T. Hao and J. Li, Backward stochastic differential equations coupled with value function and related optimal control problems, Abstract Appl. Anal., 2014 (2014), Art. ID 262713, 17 pp. doi: 10.1155/2014/262713. [8] 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. [9] J. Li, Reflected mean-field backward stochastic differential equations. Approximation and associated nonlinear PDEs, J. Math. Anal. Appl., 413 (2014), 47-68. doi: 10.1016/j.jmaa.2013.11.028. [10] Z. Li and J. Luo, Mean-field reflected backward stochastic differential equations, Stat. Probab. Lett., 82 (2012), 1961-1968. doi: 10.1016/j.spl.2012.06.018. [11] J. Yan, S. Peng and S. Fang, BSDE and stochastic optimizations, in Topics in Stochastic Analysis (eds. L. Wu), Science Press, 1997 (In Chinese).

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##### References:
 [1] G. Barles, R. Buckdahn and E. Pardoux, Backward stochastic differential equations and integral-partial differential equations, Stoch. Stoch. Rep., 60 (1997), 57-83. doi: 10.1080/17442509708834099. [2] R. Buckdahn and J. Li, Stochastic differential games and viscosity solutions of Hamilton- Jacobi-Bellman-Isaacs equations, SIAM. J. Control. Optim., 47 (2008), 444-475. doi: 10.1137/060671954. [3] R. Buckdahn and J. Li, Stochastic differential games with reflection and related obstacle problems for Isaacs equations, Acta Math. Appl. Sin.-Enql. Ser., 27 (2011), 647-678. doi: 10.1007/s10255-011-0068-8. [4] R. Buckdahn, B. Djehiche, J. Li and S. Peng, Mean-field backward stochastic differential equations. A limit approach, Ann. Probab., 37 (2009), 1524-1565. doi: 10.1214/08-AOP442. [5] R. Buckdahn, J. Li and S. Peng, Mean-field backward stochastic differential equations and related partial differential equations, Stoch. Proc. App., 119 (2009), 3133-3154. doi: 10.1016/j.spa.2009.05.002. [6] M. G. Crandall, H. Ishii and P. L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc., 27 (1992), 1-67. doi: 10.1090/S0273-0979-1992-00266-5. [7] T. Hao and J. Li, Backward stochastic differential equations coupled with value function and related optimal control problems, Abstract Appl. Anal., 2014 (2014), Art. ID 262713, 17 pp. doi: 10.1155/2014/262713. [8] 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. [9] J. Li, Reflected mean-field backward stochastic differential equations. Approximation and associated nonlinear PDEs, J. Math. Anal. Appl., 413 (2014), 47-68. doi: 10.1016/j.jmaa.2013.11.028. [10] Z. Li and J. Luo, Mean-field reflected backward stochastic differential equations, Stat. Probab. Lett., 82 (2012), 1961-1968. doi: 10.1016/j.spl.2012.06.018. [11] J. Yan, S. Peng and S. Fang, BSDE and stochastic optimizations, in Topics in Stochastic Analysis (eds. L. Wu), Science Press, 1997 (In Chinese).
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