$L^{p}$ SOLUTION OF GENERAL MEAN-FIELD BSDES WITH CONTINUOUS COEFFICIENTS

doi:10.1007/s10473-020-0417-x

Abstract

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 $P_{Y}$ of $Y$ . The first part of the paper is devoted to the existence and uniqueness of solutions in $L^p$ , $1< p\leq2$ , where the monotonicity conditions are satisfied. Next, we show that if the generator $f$ is uniformly continuous in $(\mu,y,z)$ , uniformly with respect to $(t,\omega)$ , and if the terminal value $\xi$ belongs to $L^{p}(\Omega,\mathcal{F},P)$ with $1

Key words: general mean-field backward stochastic differential equations, monotonicity condition, continuous condition, uniformly continuous condition, L^{p

CLC Number:

60H10

Yajie CHEN, Chuanzhi XING, Xiao ZHANG. $L^{p}$ SOLUTION OF GENERAL MEAN-FIELD BSDES WITH CONTINUOUS COEFFICIENTS[J].Acta mathematica scientia,Series B, 2020, 40(4): 1116-1140.

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References

[1] Briand Ph, Delyon B, Hu Y, et al.

$L^{p}$ solutions of backward stochastic differential equations. Stochastic Processes and their Applications, 2003, 108(1):109-129
[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
[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
[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
[5] Cardaliaguet P. Notes on Mean Field Games. (from P.-L. Lions' lectures at Collège de France), 2013
[6] Chen S.

$L^{p}$ solutions of one-dimensional backward stochastic differential equations with continuous coefficients. Stochastic Analysis and their Applications, 2010, 28(5):820-841
[7] Karoui N El, Peng S, Quenez M C. Backward stochastic differential equations in finance. Mathematical Finance, 1997, 7(1):1-71
[8] Fan S, Jiang L, Davison M. Uniqueness of solutions for multidimensional BSDEs with uniformly continuous generators. C R Acad, 2010, 348:683-686
[9] Hamadène S. Multidimensional backward stochastic differential equations with uniformly continuous coefficients. Bernoulli, 2003, 9(3):517-534
[10] Jia G. A class of backward stochastic differential equations with discontinuous coefficients. Statistics and Probability, 2008, 78(3):231-237
[11] Jia G. On existence of backward stochastic differential equations with left-Lipschitz coefficient. Chinese Annal of Mathematics, 2007, 5(5):601-610
[12] Kac M. Foundations of kinetic theory. Berkeley Symposium on Mathematical Statistics and Probability, 1956, 3(69):101-110
[13] Lasry J M, Lions P L. Mean filed games. Japan J Math, 2007, 2:229-260
[14] Lepeltier J P, San Martin J. Backward stochastic differential equations with continuous coefficient. Statistics and Probability, 1997, 32(4):425-430
[15] 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
[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
[17] Lions P L. Cours au Collège de France:Théorie des jeu à champs moyens. 2013, http://www.college-defrance.fr
[18] Pardoux E, Peng S. Adapted solution of a backward stochastic differential equation. Systems & Control Letters, 1990, 14(1):55-61
[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
[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
[21] Peng S. Probabilistic interpretation for systems of quasilinear parabolic partial differential equations. Stochastics and Stochastics Reports, 1991, 37(1):61-74
[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
[23] Wu Z, Xiao H. Multi-dimensional reflected backward stochastic differential equations and the comparison theorem. Acta Math Sci, 2010, 30B(5):1819-1836