doi: 10.3934/naco.2022009
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Mean-field type quadratic BSDEs

1. 

Univ. Rennes, CNRS, IRMAR-UMR6625, F-35000, Rennes, France

2. 

School of Mathematical Sciences, Fudan University, Shanghai 200433, China

3. 

Department of Finance and Control Sciences, School of Mathematical Sciences, Fudan University, Shanghai 200433, China

*Corresponding author: Ying Hu

Received  November 2021 Revised  April 2022 Early access May 2022

In this paper, we give several new results on solvability of a quadratic BSDE whose generator depends also on the mean of both variables. First, we consider such a BSDE using John-Nirenberg's inequality for BMO martingales to estimate its contribution to the evolution of the first unknown variable. Then we consider the BSDE having an additive expected value of a quadratic generator in addition to the usual quadratic one. In this case, we use a deterministic shift transformation to the first unknown variable, when the usual quadratic generator depends neither on the first variable nor its mean. The general case can be treated by a fixed point argument.

Citation: Hélène Hibon, Ying Hu, Shanjian Tang. Mean-field type quadratic BSDEs. Numerical Algebra, Control and Optimization, doi: 10.3934/naco.2022009
References:
[1]

A. BensoussanS. C. P. Yam and Z. Zhang, Well-posedness of mean-field type forward-backward stochastic differential equations, Stochastic Process. Appl., 125 (2015), 3327-3354.  doi: 10.1016/j.spa.2015.04.006.

[2]

J.-M. Bismut, Conjugate convex functions in optimal stochastic control, J. Math. Anal. Appl., 44 (1973), 384-404.  doi: 10.1016/0022-247X(73)90066-8.

[3]

J.-M. Bismut, Linear quadratic optimal stochastic control with random coefficients, SIAM J. Control Optim., 14 (1976), 419-444.  doi: 10.1137/0314028.

[4]

Ph. Briand and Y. Hu, BSDE with quadratic growth and unbounded terminal value, Probab. Theory Related Fields, 136 (2006), 604-618.  doi: 10.1007/s00440-006-0497-0.

[5]

Ph. Briand and Y. Hu, Quadratic BSDEs with convex generators and unbounded terminal conditions, Probab. Theory Related Fields, 141 (2008), 543-567.  doi: 10.1007/s00440-007-0093-y.

[6]

R. BuckdahnJ. Li and S. Peng, Mean-field backward stochastic differential equations and related partial differential equations, Stochastic Process. Appl., 119 (2009), 3133-3154.  doi: 10.1016/j.spa.2009.05.002.

[7]

R. Carmona and F. Delarue, Mean field forward-backward stochastic differential equations, Electron. Commun. Probab., 18 (2013), 15 pages. doi: 10.1214/ECP.v18-2446.

[8]

R. Carmona and F. Delarue, Probabilistic Theory of Mean Field Games With Applications, I. Mean Field Fbsdes, Control, and Games, Probability Theory and Stochastic Modelling, 83 (2018), Springer, Cham.

[9]

P. Cheridito and K. Nam, BSEs, BSDEs and fixed point problems, Ann. Probab., 45 (2017), 3795-3828.  doi: 10.1214/16-AOP1149.

[10]

N. Kazamaki, Continuous Exponential Martingales and BMO, Lecture Notes in Mathematics, 1579 (1994), Springer-Verlag, Berlin. doi: 10.1007/BFb0073585.

[11]

M. Kobylanski, Backward stochastic differential equations and partial differential equations with quadratic growth, Ann. Probab., 28 (2000), 558-602.  doi: 10.1214/aop/1019160253.

[12]

E. Pardoux and S. Peng, Adapted solution of a backward stochastic differential equation, Systems Control Lett., 14 (1990), 55-61.  doi: 10.1016/0167-6911(90)90082-6.

show all references

References:
[1]

A. BensoussanS. C. P. Yam and Z. Zhang, Well-posedness of mean-field type forward-backward stochastic differential equations, Stochastic Process. Appl., 125 (2015), 3327-3354.  doi: 10.1016/j.spa.2015.04.006.

[2]

J.-M. Bismut, Conjugate convex functions in optimal stochastic control, J. Math. Anal. Appl., 44 (1973), 384-404.  doi: 10.1016/0022-247X(73)90066-8.

[3]

J.-M. Bismut, Linear quadratic optimal stochastic control with random coefficients, SIAM J. Control Optim., 14 (1976), 419-444.  doi: 10.1137/0314028.

[4]

Ph. Briand and Y. Hu, BSDE with quadratic growth and unbounded terminal value, Probab. Theory Related Fields, 136 (2006), 604-618.  doi: 10.1007/s00440-006-0497-0.

[5]

Ph. Briand and Y. Hu, Quadratic BSDEs with convex generators and unbounded terminal conditions, Probab. Theory Related Fields, 141 (2008), 543-567.  doi: 10.1007/s00440-007-0093-y.

[6]

R. BuckdahnJ. Li and S. Peng, Mean-field backward stochastic differential equations and related partial differential equations, Stochastic Process. Appl., 119 (2009), 3133-3154.  doi: 10.1016/j.spa.2009.05.002.

[7]

R. Carmona and F. Delarue, Mean field forward-backward stochastic differential equations, Electron. Commun. Probab., 18 (2013), 15 pages. doi: 10.1214/ECP.v18-2446.

[8]

R. Carmona and F. Delarue, Probabilistic Theory of Mean Field Games With Applications, I. Mean Field Fbsdes, Control, and Games, Probability Theory and Stochastic Modelling, 83 (2018), Springer, Cham.

[9]

P. Cheridito and K. Nam, BSEs, BSDEs and fixed point problems, Ann. Probab., 45 (2017), 3795-3828.  doi: 10.1214/16-AOP1149.

[10]

N. Kazamaki, Continuous Exponential Martingales and BMO, Lecture Notes in Mathematics, 1579 (1994), Springer-Verlag, Berlin. doi: 10.1007/BFb0073585.

[11]

M. Kobylanski, Backward stochastic differential equations and partial differential equations with quadratic growth, Ann. Probab., 28 (2000), 558-602.  doi: 10.1214/aop/1019160253.

[12]

E. Pardoux and S. Peng, Adapted solution of a backward stochastic differential equation, Systems Control Lett., 14 (1990), 55-61.  doi: 10.1016/0167-6911(90)90082-6.

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