COMPARISON THEOREMS FOR MULTI-DIMENSIONAL GENERAL MEAN-FIELD BDSDES

doi:10.1007/s10473-021-0216-z

数学物理学报(英文版) ›› 2021, Vol. 41 ›› Issue (2): 535-551.doi: 10.1007/s10473-021-0216-z

COMPARISON THEOREMS FOR MULTI-DIMENSIONAL GENERAL MEAN-FIELD BDSDES

李娟¹, 邢传智¹, 彭滢²

1. School of Mathematics and Statistics, Shandong University, Weihai 264209, China;
2. Shandong University-Zhongtai Securities Institute for Financial Studies, Shandong University, Jinan 250100, China

收稿日期:2020-01-19 修回日期:2020-10-26 出版日期:2021-04-25 发布日期:2021-04-29
通讯作者: Chuanzhi XING, Ying PENG E-mail:chuanzhixing@mail.sdu.edu.cn;pengy@sdu.edu.cn
作者简介:Juan LI,E-mail:juanli@sdu.edu.cn
基金资助:
The work has been supported in part by the NSF of P.R.China (11871037; 11222110), Shandong Province (JQ201202), NSFC-RS (11661130148; NA150344), 111 Project (B12023).

COMPARISON THEOREMS FOR MULTI-DIMENSIONAL GENERAL MEAN-FIELD BDSDES

Juan LI¹, Chuanzhi XING¹, Ying PENG²

1. School of Mathematics and Statistics, Shandong University, Weihai 264209, China;
2. Shandong University-Zhongtai Securities Institute for Financial Studies, Shandong University, Jinan 250100, China

Received:2020-01-19 Revised:2020-10-26 Online:2021-04-25 Published:2021-04-29
Contact: Chuanzhi XING, Ying PENG E-mail:chuanzhixing@mail.sdu.edu.cn;pengy@sdu.edu.cn
About author:Juan LI,E-mail:juanli@sdu.edu.cn
Supported by:
The work has been supported in part by the NSF of P.R.China (11871037; 11222110), Shandong Province (JQ201202), NSFC-RS (11661130148; NA150344), 111 Project (B12023).

摘要/Abstract

摘要： 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.

关键词: Backward doubly stochastic differential equations, mean-field, multi-dimensional comparison theorem, continuous condition

Abstract: 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.

Key words: Backward doubly stochastic differential equations, mean-field, multi-dimensional comparison theorem, continuous condition

中图分类号:

60H10

李娟, 邢传智, 彭滢. COMPARISON THEOREMS FOR MULTI-DIMENSIONAL GENERAL MEAN-FIELD BDSDES[J]. 数学物理学报(英文版), 2021, 41(2): 535-551.

Juan LI, Chuanzhi XING, Ying PENG. COMPARISON THEOREMS FOR MULTI-DIMENSIONAL GENERAL MEAN-FIELD BDSDES[J]. Acta mathematica scientia,Series B, 2021, 41(2): 535-551.

参考文献

[1] Buckdahn R, Djehiche B, Li J, et al. Mean-field backward stochastic differential equations. A limit approach. Annals of Probability, 2009, 37(4):1524-1565
[2] Buckdahn R, Li J, Peng S. Mean-field backward stochastic differential equations and related partial differential equations. Stochastic Processes and their Applications, 2009, 119(10):3133-3154
[3] Buckdahn R, Li J, Peng S, et al. Mean-field stochastic differential equations and associated PDEs. Annals of Probability, 2017, 45(2):824-878
[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, 2000, 116:485-504
[5] Chen Y, Xing C, Zhang X. L^p solution of general mean-field BSDEs with continuous coefficients. Acta Mathematica Scientia, 2020, 40B(4):1116-1140
[6] Hamadène S. Multidimensional backward stochastic differential equations with uniformly continuous coefficients. Bernoulli, 2003, 9(3):517-534
[7] Hu Y, Peng S. On the comparison theorem for multidimensional BSDEs. Comptes Rendus Mathematique, 2006, 343(2):135-140
[8] Jia G. A uniqueness theorem for the solution of backward stochastic differential equations. Comptes Rendus Mathematique, 2008, 346(7/8):439-444
[9] Kac M. Foundations of kinetic theory. Proceedings of The third Berkeley symposium on mathematical statistics and probability. 1956, 3(3):171-197
[10] Lasry J M, Lions P L. Mean field games. Japan J Math, 2007, 2:229-260
[11] Lepeltier J P, San Martin J. Backward stochastic differential equations with continuous coefficient. Statistics and Probability, 1997, 32(4):425-430
[12] Li J. Mean-field forward and backward SDEs with jumps and associated nonlocal quasi-linear integralPDEs. Stochastic Processes and their Applications, 2018, 128(9):3118-3180
[13] Li J, Liang H, Zhang X. General mean-field BSDEs with continuous coefficients. Journal of Mathematical Analysis and Applications, 2018, 466(1):264-280
[14] Li J, Xing C. General mean-field BDSDEs with continuous coefficients. 2020, Submitted
[15] Lin Q, Wu Z. A comparison theorem and uniqueness theorem of backward doubly stochastic differential equations. Acta Mathematicae Applicatae Sinica, English Series, 2011, 27(2):223-232
[16] Pardoux E, Peng S. Adapted solution of a backward stochastic differential equation. Systems&Control Letters, 1990, 14(1):55-61
[17] Pardoux E, Peng S. Backward doubly stochastic differential equations and systems of quasilinear SPDEs. Probab Theory Relat Fields, 1994, 98:209-227
[18] Shi Y, Gu Y, Liu K. Comparison theorem of backward doubly stochastic differential equations and applications. Stochastic Analysis and Application, 2005, 23:97-110
[19] Zhu B, Han B. Comparison theorems for the multidimensional BDSDEs and applications. Journal of Applied Mathematics, 2012, 2012

COMPARISON THEOREMS FOR MULTI-DIMENSIONAL GENERAL MEAN-FIELD BDSDES

COMPARISON THEOREMS FOR MULTI-DIMENSIONAL GENERAL MEAN-FIELD BDSDES

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