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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

Juan Li 1, and  Wenqiang Li 1,

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.
Keywords: Reflected mean-field backward stochastic differential equations (reflected MFBSDEs), dynamic programming principle, viscosity solution., comparison theorem, reflected MFBSDEs coupled with value function.
Mathematics Subject Classification: Primary: 60H10, 60H30; Secondary: 90C39, 93E2.
Citation: Juan Li, Wenqiang Li. Controlled reflected mean-field backward stochastic differential equations coupled with value function and related PDEs. Mathematical Control & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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). Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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). Google Scholar

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