# American Institute of Mathematical Sciences

• Previous Article
A two-stage data envelopment analysis approach to solve extended transportation problem with non-homogenous costs
• NACO Home
• This Issue
• Next Article
Geodesic $\mathcal{E}$-prequasi-invex function and its applications to non-linear programming problems
doi: 10.3934/naco.2022012
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

## 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. Bahlali, S. 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. Briand, R. Elie and Y. Hu, BSDEs with mean reflection, The Annals of Applied Probability, 28 (2018), 482-510.  doi: 10.1214/17-AAP1310. [3] R. Buckdahn, B. Djehiche, J. 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. Buckdahn, J. 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. Buckdahn, J. Li, S. 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 Asri, S. 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 Karoui, C. Kapoudjian, E. Pardoux, S. 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. Li, H. 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. Bahlali, S. 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. Briand, R. Elie and Y. Hu, BSDEs with mean reflection, The Annals of Applied Probability, 28 (2018), 482-510.  doi: 10.1214/17-AAP1310. [3] R. Buckdahn, B. Djehiche, J. 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. Buckdahn, J. 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. Buckdahn, J. Li, S. 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 Asri, S. 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 Karoui, C. Kapoudjian, E. Pardoux, S. 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. Li, H. 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.
 [1] Juan Li, Wenqiang Li. Controlled reflected mean-field backward stochastic differential equations coupled with value function and related PDEs. Mathematical Control and Related Fields, 2015, 5 (3) : 501-516. doi: 10.3934/mcrf.2015.5.501 [2] René Aïd, Roxana Dumitrescu, Peter Tankov. The entry and exit game in the electricity markets: A mean-field game approach. Journal of Dynamics and Games, 2021, 8 (4) : 331-358. doi: 10.3934/jdg.2021012 [3] Yufeng Shi, Tianxiao Wang, Jiongmin Yong. Mean-field backward stochastic Volterra integral equations. Discrete and Continuous Dynamical Systems - B, 2013, 18 (7) : 1929-1967. doi: 10.3934/dcdsb.2013.18.1929 [4] Kai Du, Jianhui Huang, Zhen Wu. Linear quadratic mean-field-game of backward stochastic differential systems. Mathematical Control and Related Fields, 2018, 8 (3&4) : 653-678. doi: 10.3934/mcrf.2018028 [5] Stamatios Katsikas, Vassilli Kolokoltsov. Evolutionary, mean-field and pressure-resistance game modelling of networks security. Journal of Dynamics and Games, 2019, 6 (4) : 315-335. doi: 10.3934/jdg.2019021 [6] Jianhui Huang, Shujun Wang, Zhen Wu. Backward-forward linear-quadratic mean-field games with major and minor agents. Probability, Uncertainty and Quantitative Risk, 2016, 1 (0) : 8-. doi: 10.1186/s41546-016-0009-9 [7] Monia Karouf. Reflected solutions of backward doubly SDEs driven by Brownian motion and Poisson random measure. Discrete and Continuous Dynamical Systems, 2019, 39 (10) : 5571-5601. doi: 10.3934/dcds.2019245 [8] Ying Hu, Shanjian Tang. Switching game of backward stochastic differential equations and associated system of obliquely reflected backward stochastic differential equations. Discrete and Continuous Dynamical Systems, 2015, 35 (11) : 5447-5465. doi: 10.3934/dcds.2015.35.5447 [9] Adel Chala, Dahbia Hafayed. On stochastic maximum principle for risk-sensitive of fully coupled forward-backward stochastic control of mean-field type with application. Evolution Equations and Control Theory, 2020, 9 (3) : 817-843. doi: 10.3934/eect.2020035 [10] Hyeng Keun Koo, Shanjian Tang, Zhou Yang. A Dynkin game under Knightian uncertainty. Discrete and Continuous Dynamical Systems, 2015, 35 (11) : 5467-5498. doi: 10.3934/dcds.2015.35.5467 [11] Patrick Gerard, Christophe Pallard. A mean-field toy model for resonant transport. Kinetic and Related Models, 2010, 3 (2) : 299-309. doi: 10.3934/krm.2010.3.299 [12] Thierry Paul, Mario Pulvirenti. Asymptotic expansion of the mean-field approximation. Discrete and Continuous Dynamical Systems, 2019, 39 (4) : 1891-1921. doi: 10.3934/dcds.2019080 [13] Seung-Yeal Ha, Jinwook Jung, Jeongho Kim, Jinyeong Park, Xiongtao Zhang. A mean-field limit of the particle swarmalator model. Kinetic and Related Models, 2021, 14 (3) : 429-468. doi: 10.3934/krm.2021011 [14] Marco Cirant, Diogo A. Gomes, Edgard A. Pimentel, Héctor Sánchez-Morgado. On some singular mean-field games. Journal of Dynamics and Games, 2021, 8 (4) : 445-465. doi: 10.3934/jdg.2021006 [15] Hélène Hibon, Ying Hu, Shanjian Tang. Mean-field type quadratic BSDEs. Numerical Algebra, Control and Optimization, 2022  doi: 10.3934/naco.2022009 [16] César Barilla, Guillaume Carlier, Jean-Michel Lasry. A mean field game model for the evolution of cities. Journal of Dynamics and Games, 2021, 8 (3) : 299-329. doi: 10.3934/jdg.2021017 [17] Illés Horváth, Kristóf Attila Horváth, Péter Kovács, Miklós Telek. Mean-field analysis of a scaling MAC radio protocol. Journal of Industrial and Management Optimization, 2021, 17 (1) : 279-297. doi: 10.3934/jimo.2019111 [18] Diogo A. Gomes, Gabriel E. Pires, Héctor Sánchez-Morgado. A-priori estimates for stationary mean-field games. Networks and Heterogeneous Media, 2012, 7 (2) : 303-314. doi: 10.3934/nhm.2012.7.303 [19] Gerasimenko Viktor. Heisenberg picture of quantum kinetic evolution in mean-field limit. Kinetic and Related Models, 2011, 4 (1) : 385-399. doi: 10.3934/krm.2011.4.385 [20] Franco Flandoli, Enrico Priola, Giovanni Zanco. A mean-field model with discontinuous coefficients for neurons with spatial interaction. Discrete and Continuous Dynamical Systems, 2019, 39 (6) : 3037-3067. doi: 10.3934/dcds.2019126

Impact Factor: