Comparison theorems for multi-dimensional general mean-field BDSDES. (English) Zbl 1513.60072
Summary: In this paper we study multi-dimensional mean-field backward doubly stochastic differential equations (BDSDEs), that is, BDSDEs whose coefficients depend not only on the solution processes but also on their law. The first part of the paper is devoted to the comparison theorem for multi-dimensional mean-field BDSDEs with Lipschitz conditions. With the help of the comparison result for the Lipschitz case we prove the existence of a solution for multi-dimensional mean-field BDSDEs with an only continuous drift coefficient of linear growth, and we also extend the comparison theorem to such BDSDEs with a continuous coefficient.
MSC:
| 60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |
Keywords:
backward doubly stochastic differential equations; mean-field; multi-dimensional comparison theorem; continuous conditionReferences:
| [1] | Buckdahn, R.; Djehiche, B.; Li, J., Mean-field backward stochastic differential equations, A limit approach. Annals of Probability, 37, 4, 1524-1565 (2009) · Zbl 1176.60042 |
| [2] | Buckdahn, R.; Li, J.; Peng, S., Mean-field backward stochastic differential equations and related partial differential equations, Stochastic Processes and their Applications, 119, 10, 3133-3154 (2009) · Zbl 1183.60022 · doi:10.1016/j.spa.2009.05.002 |
| [3] | Buckdahn, R.; Li, J.; Peng, S., Mean-field stochastic differential equations and associated PDEs, Annals of Probability, 45, 2, 824-878 (2017) · Zbl 1402.60070 · doi:10.1214/15-AOP1076 |
| [4] | Buckdahn, R.; Quincampoix, M.; Rascanu, A., Viability property for a backward stochastic differential equation and applications to partial differential equations, Probab Theory Relat Fields, 116, 485-504 (2000) · Zbl 0969.60061 · doi:10.1007/s004400050260 |
| [5] | Chen, Y.; Xing, C.; Zhang, X., L^p solution of general mean-field BSDEs with continuous coefficients, Acta Mathematica Scientia, 40B, 4, 1116-1140 (2020) · Zbl 1499.60180 · doi:10.1007/s10473-020-0417-x |
| [6] | Hamadène, S., Multidimensional backward stochastic differential equations with uniformly continuous coefficients, Bernoulli, 9, 3, 517-534 (2003) · Zbl 1048.60047 · doi:10.3150/bj/1065444816 |
| [7] | Hu, Y.; Peng, S., On the comparison theorem for multidimensional BSDEs, Comptes Rendus Mathematique, 343, 2, 135-140 (2006) · Zbl 1098.60052 · doi:10.1016/j.crma.2006.05.019 |
| [8] | Jia, G., A uniqueness theorem for the solution of backward stochastic differential equations, Comptes Rendus Mathematique, 346, 7-8, 439-444 (2008) · Zbl 1143.60040 · doi:10.1016/j.crma.2008.02.012 |
| [9] | Kac, M., Foundations of kinetic theory, Proceedings of The third Berkeley symposium on mathematical statistics and probability, 3, 3, 171-197 (1956) · Zbl 0072.42802 |
| [10] | Lasry, J. M.; Lions, P. L., Mean field games, Japan J Math, 2, 229-260 (2007) · Zbl 1156.91321 · doi:10.1007/s11537-007-0657-8 |
| [11] | Lepeltier, J. P.; San Martin, J., Backward stochastic differential equations with continuous coefficient, Statistics and Probability, 32, 4, 425-430 (1997) · Zbl 0904.60042 · doi:10.1016/S0167-7152(96)00103-4 |
| [12] | Li, J., Mean-field forward and backward SDEs with jumps and associated nonlocal quasi-linear integral-PDEs, Stochastic Processes and their Applications, 128, 9, 3118-3180 (2018) · Zbl 1405.60078 · doi:10.1016/j.spa.2017.10.011 |
| [13] | Li, J.; Liang, H.; Zhang, X., General mean-field BSDEs with continuous coefficients, Journal of Mathematical Analysis and Applications, 466, 1, 264-280 (2018) · Zbl 1390.60214 · doi:10.1016/j.jmaa.2018.05.074 |
| [14] | Li J, Xing C. General mean-field BDSDEs with continuous coefficients. 2020, Submitted · Zbl 1481.60111 |
| [15] | Lin, Q.; Wu, Z., A comparison theorem and uniqueness theorem of backward doubly stochastic differential equations, Acta Mathematicae Applicatae Sinica, English Series, 27, 2, 223-232 (2011) · Zbl 1216.60053 · doi:10.1007/s10255-011-0057-y |
| [16] | Pardoux, E.; Peng, S., Adapted solution of a backward stochastic differential equation, Systems&Control Letters, 14, 1, 55-61 (1990) · Zbl 0692.93064 |
| [17] | Pardoux, E.; Peng, S., Backward doubly stochastic differential equations and systems of quasilinear SPDEs, Probab Theory Relat Fields, 98, 209-227 (1994) · Zbl 0792.60050 · doi:10.1007/BF01192514 |
| [18] | Shi, Y.; Gu, Y.; Liu, K., Comparison theorem of backward doubly stochastic differential equations and applications, Stochastic Analysis and Application, 23, 97-110 (2005) · Zbl 1067.60046 · doi:10.1081/SAP-200044444 |
| [19] | Zhu B, Han B. Comparison theorems for the multidimensional BDSDEs and applications. Journal of Applied Mathematics, 2012, 2012 · Zbl 1244.60064 |
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