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Lp solution of general mean-field BSDEs with continuous coefficients

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Abstract

In this paper we consider one dimensional mean-field backward stochastic differential equations (BSDEs) under weak assumptions on the coefficient. Unlike [3], the generator of our mean-field BSDEs depends not only on the solution (Y, Z) but also on the law PY of Y. The first part of the paper is devoted to the existence and uniqueness of solutions in Lp, 1 < p ≤ 2, where the monotonicity conditions are satisfied. Next, we show that if the generator f is uniformly continuous in (μ, y, z), uniformly with respect to (t, ω), and if the terminal value ξ belongs to Lp(Ω, F, P) with 1 < p ≤ 2, the mean-field BSDE has a unique Lp solution.

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References

  1. Briand Ph, Delyon B, Hu Y, et al. Lp solutions of backward stochastic differential equations. Stochastic Processes and their Applications, 2003, 108(1): 109–129

    Article  MathSciNet  Google Scholar 

  2. 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

    MathSciNet  MATH  Google Scholar 

  3. 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

    Article  MathSciNet  Google Scholar 

  4. Buckdahn R, Li J, Peng S, et al. Mean-field stochastic differential equations and associated PDEs. The Annals of Probability, 2017, 45(2): 824–878

    Article  MathSciNet  Google Scholar 

  5. Cardaliaguet P. Notes on Mean Field Games. (from P.-L. Lions’ lectures at Collège de France), 2013

  6. Chen S. Lp solutions of one-dimensional backward stochastic differential equations with continuous coefficients. Stochastic Analysis and their Applications, 2010, 28(5): 820–841

    Article  Google Scholar 

  7. Karoui N El, Peng S, Quenez M C. Backward stochastic differential equations in finance. Mathematical Finance, 1997, 7(1): 1–71

    Article  MathSciNet  Google Scholar 

  8. Fan S, Jiang L, Davison M. Uniqueness of solutions for multidimensional BSDEs with uniformly continuous generators. C R Acad, 2010, 348: 683–686

    MathSciNet  MATH  Google Scholar 

  9. Hamadène S. Multidimensional backward stochastic differential equations with uniformly continuous coefficients. Bernoulli, 2003, 9(3): 517–534

    Article  MathSciNet  Google Scholar 

  10. Jia G. A class of backward stochastic differential equations with discontinuous coefficients. Statistics and Probability, 2008, 78(3): 231–237

    Article  MathSciNet  Google Scholar 

  11. Jia G. On existence of backward stochastic differential equations with left-Lipschitz coefficient. Chinese Annal of Mathematics, 2007, 5(5): 601–610

    MathSciNet  MATH  Google Scholar 

  12. Kac M. Foundations of kinetic theory. Berkeley Symposium on Mathematical Statistics and Probability, 1956, 3(69): 101–110

    MathSciNet  MATH  Google Scholar 

  13. Lasry J M, Lions P L. Mean filed games. Japan J Math, 2007, 2: 229–260

    Article  Google Scholar 

  14. Lepeltier J P, San Martin J. Backward stochastic differential equations with continuous coefficient. Statistics and Probability, 1997, 32(4): 425–430

    Article  MathSciNet  Google Scholar 

  15. Li J. Mean-field forward and backward SDEs with jumps and associated nonlocal quasi-linear integral-PDEs. Stochastic Processes and their Applications, 2018, 128(9): 3118–3180

    Article  MathSciNet  Google Scholar 

  16. Li J, Liang H, Zhang X. General mean-field BSDEs with continuous coefficients. Journal of Mathematical Analysis and Applications, 2018, 466(1): 264–280

    Article  MathSciNet  Google Scholar 

  17. Lions P L. Cours au Collège de France: Théorie des jeu à champs moyens. 2013, http://www.college-de-france.fr

  18. Pardoux E, Peng S. Adapted solution of a backward stochastic differential equation. Systems & Control Letters, 1990, 14(1): 55–61

    Article  MathSciNet  Google Scholar 

  19. Pardoux E, Peng S. Backward stochastic differential equations and quasilinear parabolic partial differential equations. Stochastic Partial Differential Equations and Their Applications, 1992, 176(1): 200–217

    Article  MathSciNet  Google Scholar 

  20. Peng S. A generalized dynamic programming principle and Hamilton-Jacobi-Bellman equation. Stochastics An International Journal of Probability&Stochastic Processes, 1992, 38(2): 119–134

    MathSciNet  MATH  Google Scholar 

  21. Peng S. Probabilistic interpretation for systems of quasilinear parabolic partial differential equations. Stochastics and Stochastics Reports, 1991, 37(1): 61–74

    MathSciNet  MATH  Google Scholar 

  22. Tian D, Jiang L, Davison M. On the existence of solutions to BSDEs with generalized uniformly continuous generators. Statistics and Probability, 2010, 80(9): 903–909

    Article  MathSciNet  Google Scholar 

  23. Wu Z, Xiao H. Multi-dimensional reflected backward stochastic differential equations and the comparison theorem. Acta Math Sci, 2010, 30B(5): 1819–1836

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Zhongtai Securities Institute for Financial Studies, Shandong University, Jinan, 250100, China

    Yajie Chen

  2. School of Mathematics and Statistics, Shandong University, Weihai, 264209, China

    Chuanzhi Xing & Xiao Zhang

Authors
  1. Yajie Chen
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  2. Chuanzhi Xing
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  3. Xiao Zhang
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Corresponding author

Correspondence to Chuanzhi Xing.

Additional information

The work was supported in part by the NSFC (11222110; 11871037), Shandong Province (JQ201202), NSFC-RS (11661130148; NA150344), 111 Project (B12023).

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Chen, Y., Xing, C. & Zhang, X. Lp solution of general mean-field BSDEs with continuous coefficients. Acta Math Sci 40, 1116–1140 (2020). https://doi.org/10.1007/s10473-020-0417-x

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  • DOI: https://doi.org/10.1007/s10473-020-0417-x

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