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doi: 10.3934/naco.2022012
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Mean-field doubly reflected backward stochastic differential equations

1. 

Department of Mathematics, Shandong University, Jinan, Shandong Province, China

2. 

Le Mans University, LMM, Avenue Olivier Messiaen, 72085 Le Mans, Cedex 9, France

This paper is dedicated to Professor Jin Ma on the occasion of his 65-th Birthday.

Received  March 2022 Revised  April 2022 Early access May 2022

We study mean-field doubly reflected BSDEs. First, using the fixed point method, we show existence and uniqueness of the solution when the data which define the BSDE are $ p $-integrable with $ p = 1 $ or $ p>1 $. The two cases are treated separately. Next by penalization we show also the existence of the solution. The two methods do not cover the same set of assumptions.

Citation: Yinggu Chen, Said HamadÈne, Tingshu Mu. Mean-field doubly reflected backward stochastic differential equations. Numerical Algebra, Control and Optimization, doi: 10.3934/naco.2022012
References:
[1]

K. BahlaliS. Hamadene and B. Mezerdi, Backward stochastic differential equations with two reflecting barriers and continuous with quadratic growth coefficient, Stochastic Processes and Their Applications, 115 (2005), 1107-1129.  doi: 10.1016/j.spa.2005.02.005.

[2]

P. BriandR. Elie and Y. Hu, BSDEs with mean reflection, The Annals of Applied Probability, 28 (2018), 482-510.  doi: 10.1214/17-AAP1310.

[3]

R. BuckdahnB. DjehicheJ. Li and S. Peng, Mean-field backward stochastic differential equations: a limit approach, The Annals of Probability, 37 (2009), 1524-1565.  doi: 10.1214/08-AOP442.

[4]

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

[5]

R. BuckdahnJ. LiS. Peng and C. Rainer, Mean-field stochastic differential equations and associated PDEs, The Annals of Probability, 45 (2017), 824-878.  doi: 10.1214/15-AOP1076.

[6]

R. Carmona and F. Delarue, Mean field forward-backward stochastic differential equations, Electronic Communications in Probability, 18 (2013). doi: 10.1214/ECP.v18-2446.

[7]

J. Cvitanic and I. Karatzas, Backward stochastic differential equations with reflection and Dynkin games, The Annals of Probability, (1996), 2024–2056. doi: 10.1214/aop/1041903216.

[8]

C. Dellacherie and P. A. Meyer, Probabilities et potentiel. Chapitres V VIII, revised ed. Actualits Scientifiques et Industrielles [Current Scientific and Industrial Topics], (1980). 1385.

[9]

B. Djehiche and R. Dumitrescu, Zero-sum mean-field Dynkin games: characterization and convergence, arXiv: 2202.02126, 2022.

[10]

B. Djehiche, R. Dumitrescu and J. Zeng, A propagation of chaos result for a class of mean-field reflected BSDEs with jumps, arXiv: 2111.14315, 2021.

[11]

B. Djehiche and R. Elie and S. Hamadéne, Mean-field reflected backward stochastic differential equations, arXiv: 1911.06079, To appear in Annals of Applied Probability, 2019.

[12]

B. El AsriS. Hamadne and H. Wang, $L^p$-solutions for doubly reflected backward stochastic differential equations, Stochastic Analysis and Applications, 29 (2011), 907-932.  doi: 10.1080/07362994.2011.564442.

[13]

N. El KarouiC. KapoudjianE. PardouxS. Peng and M.-C. Quenez, Reflected solutions of backward SDE's, and related obstacle problems for PDE's, The Annals of Probability, 25 (1997), 702-737.  doi: 10.1214/aop/1024404416.

[14]

S. Hamadéne and J. P. Lepeltier, Reflected BSDEs and mixed game problem, Stochastic Processes and Their Applications, 85 (2000), 177-188.  doi: 10.1016/S0304-4149(99)00072-1.

[15]

I. Hassairi, Existence and uniqueness for $\mathbb {D} $-solutions of reflected BSDEs with two barriers without Mokobodzki's condition, Communications on Pure & Applied Analysis, 15 (2016), 1139.  doi: 10.3934/cpaa.2016.15.1139.

[16]

Y. Hu, R. Moreau and F. Wang, Mean-Field Reflected BSDEs: the general Lipschitz case, arXiv: 2201.10359, 2022.

[17]

A. N. Kolmogorov and S. V. Fomin, Introductory Real Analysis, Courier Corporation, 1975.

[18]

J. M. Lasry and P. L. Lions, Mean field games, Japanese Journal of Mathematics, 2 (2007), 229-260.  doi: 10.1007/s11537-007-0657-8.

[19]

J. P. Lepeltier and M. A. Maingueneau, Le jeu de Dynkin en thorie gnrale sans l'hypothse de Mokobodski, Stochastics: An International Journal of Probability and Stochastic Processes, 13 (1984), 25-44.  doi: 10.1080/17442508408833309.

[20]

J. P. Lepeltier and J. San Martin, Backward SDEs with two barriers and continuous coefficient: an existence result, Journal of Applied Probability, 41 (2004), 162-175.  doi: 10.1239/jap/1077134675.

[21]

J. P. Lepeltier and M. Xu, Penalization method for reflected backward stochastic differential equations with one rcll barrier, Statistics & Probability Letters, 75 (2005), 58-66.  doi: 10.1016/j.spl.2005.05.016.

[22]

J. LiH. Liang and X. Zhang, General mean-field BSDEs with continuous coefficients, Journal of Mathematical Analysis and Applications, 466 (2018), 264-280.  doi: 10.1016/j.jmaa.2018.05.074.

[23]

J. Li, Reflected mean-field backward stochastic differential equations. Approximation and associated nonlinear PDEs, Journal of Mathematical Analysis and Applications, 413 (2014), 47-68.  doi: 10.1016/j.jmaa.2013.11.028.

[24]

E. Miller and H. Pham, Linear-quadratic McKean-Vlasov stochastic differential games, In Modeling, Stochastic Control, Optimization, and Applications, (2019), 451–481.

[25]

H. Pham, Linear quadratic optimal control of conditional McKean-Vlasov equation with random coefficients and applications, Probability, Uncertainty and Quantitative Risk, 1 (2016), 7.  doi: 10.1186/s41546-016-0008-x.

[26]

D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, Springer Science & Business Media, 293 (2013). doi: 10.1007/978-3-642-31898-6.

[27]

M. Topolewski, Reflected BSDEs with general filtration and two completely separated barriers, Probab. Math. Statist., 39 (2019), 199-218.  doi: 10.19195/0208-4147.39.1.13.

[28]

A. Uppman, Un theoreme de Helly pour les surmartingales fortes, In Sminaire de Probabilits XVI, Springer, Berlin, Heidelberg, (1980/81), 285–297.

show all references

References:
[1]

K. BahlaliS. Hamadene and B. Mezerdi, Backward stochastic differential equations with two reflecting barriers and continuous with quadratic growth coefficient, Stochastic Processes and Their Applications, 115 (2005), 1107-1129.  doi: 10.1016/j.spa.2005.02.005.

[2]

P. BriandR. Elie and Y. Hu, BSDEs with mean reflection, The Annals of Applied Probability, 28 (2018), 482-510.  doi: 10.1214/17-AAP1310.

[3]

R. BuckdahnB. DjehicheJ. Li and S. Peng, Mean-field backward stochastic differential equations: a limit approach, The Annals of Probability, 37 (2009), 1524-1565.  doi: 10.1214/08-AOP442.

[4]

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

[5]

R. BuckdahnJ. LiS. Peng and C. Rainer, Mean-field stochastic differential equations and associated PDEs, The Annals of Probability, 45 (2017), 824-878.  doi: 10.1214/15-AOP1076.

[6]

R. Carmona and F. Delarue, Mean field forward-backward stochastic differential equations, Electronic Communications in Probability, 18 (2013). doi: 10.1214/ECP.v18-2446.

[7]

J. Cvitanic and I. Karatzas, Backward stochastic differential equations with reflection and Dynkin games, The Annals of Probability, (1996), 2024–2056. doi: 10.1214/aop/1041903216.

[8]

C. Dellacherie and P. A. Meyer, Probabilities et potentiel. Chapitres V VIII, revised ed. Actualits Scientifiques et Industrielles [Current Scientific and Industrial Topics], (1980). 1385.

[9]

B. Djehiche and R. Dumitrescu, Zero-sum mean-field Dynkin games: characterization and convergence, arXiv: 2202.02126, 2022.

[10]

B. Djehiche, R. Dumitrescu and J. Zeng, A propagation of chaos result for a class of mean-field reflected BSDEs with jumps, arXiv: 2111.14315, 2021.

[11]

B. Djehiche and R. Elie and S. Hamadéne, Mean-field reflected backward stochastic differential equations, arXiv: 1911.06079, To appear in Annals of Applied Probability, 2019.

[12]

B. El AsriS. Hamadne and H. Wang, $L^p$-solutions for doubly reflected backward stochastic differential equations, Stochastic Analysis and Applications, 29 (2011), 907-932.  doi: 10.1080/07362994.2011.564442.

[13]

N. El KarouiC. KapoudjianE. PardouxS. Peng and M.-C. Quenez, Reflected solutions of backward SDE's, and related obstacle problems for PDE's, The Annals of Probability, 25 (1997), 702-737.  doi: 10.1214/aop/1024404416.

[14]

S. Hamadéne and J. P. Lepeltier, Reflected BSDEs and mixed game problem, Stochastic Processes and Their Applications, 85 (2000), 177-188.  doi: 10.1016/S0304-4149(99)00072-1.

[15]

I. Hassairi, Existence and uniqueness for $\mathbb {D} $-solutions of reflected BSDEs with two barriers without Mokobodzki's condition, Communications on Pure & Applied Analysis, 15 (2016), 1139.  doi: 10.3934/cpaa.2016.15.1139.

[16]

Y. Hu, R. Moreau and F. Wang, Mean-Field Reflected BSDEs: the general Lipschitz case, arXiv: 2201.10359, 2022.

[17]

A. N. Kolmogorov and S. V. Fomin, Introductory Real Analysis, Courier Corporation, 1975.

[18]

J. M. Lasry and P. L. Lions, Mean field games, Japanese Journal of Mathematics, 2 (2007), 229-260.  doi: 10.1007/s11537-007-0657-8.

[19]

J. P. Lepeltier and M. A. Maingueneau, Le jeu de Dynkin en thorie gnrale sans l'hypothse de Mokobodski, Stochastics: An International Journal of Probability and Stochastic Processes, 13 (1984), 25-44.  doi: 10.1080/17442508408833309.

[20]

J. P. Lepeltier and J. San Martin, Backward SDEs with two barriers and continuous coefficient: an existence result, Journal of Applied Probability, 41 (2004), 162-175.  doi: 10.1239/jap/1077134675.

[21]

J. P. Lepeltier and M. Xu, Penalization method for reflected backward stochastic differential equations with one rcll barrier, Statistics & Probability Letters, 75 (2005), 58-66.  doi: 10.1016/j.spl.2005.05.016.

[22]

J. LiH. Liang and X. Zhang, General mean-field BSDEs with continuous coefficients, Journal of Mathematical Analysis and Applications, 466 (2018), 264-280.  doi: 10.1016/j.jmaa.2018.05.074.

[23]

J. Li, Reflected mean-field backward stochastic differential equations. Approximation and associated nonlinear PDEs, Journal of Mathematical Analysis and Applications, 413 (2014), 47-68.  doi: 10.1016/j.jmaa.2013.11.028.

[24]

E. Miller and H. Pham, Linear-quadratic McKean-Vlasov stochastic differential games, In Modeling, Stochastic Control, Optimization, and Applications, (2019), 451–481.

[25]

H. Pham, Linear quadratic optimal control of conditional McKean-Vlasov equation with random coefficients and applications, Probability, Uncertainty and Quantitative Risk, 1 (2016), 7.  doi: 10.1186/s41546-016-0008-x.

[26]

D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, Springer Science & Business Media, 293 (2013). doi: 10.1007/978-3-642-31898-6.

[27]

M. Topolewski, Reflected BSDEs with general filtration and two completely separated barriers, Probab. Math. Statist., 39 (2019), 199-218.  doi: 10.19195/0208-4147.39.1.13.

[28]

A. Uppman, Un theoreme de Helly pour les surmartingales fortes, In Sminaire de Probabilits XVI, Springer, Berlin, Heidelberg, (1980/81), 285–297.

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