Numerical Resolution of McKean-Vlasov FBSDEs Using Neural Networks
We propose several algorithms to solve McKean-Vlasov Forward Backward Stochastic Differential Equations (FBSDEs). Our schemes rely on the approximating power of neural networks to estimate the solution or its gradient through minimization problems. As a consequence, we obtain methods able to tackle...
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| Vydáno v: | Methodology and computing in applied probability Ročník 24; číslo 4; s. 2557 - 2586 |
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| Médium: | Journal Article |
| Jazyk: | angličtina |
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01.12.2022
Springer Nature B.V Springer Verlag |
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| ISSN: | 1387-5841, 1573-7713 |
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| Abstract | We propose several algorithms to solve McKean-Vlasov Forward Backward Stochastic Differential Equations (FBSDEs). Our schemes rely on the approximating power of neural networks to estimate the solution or its gradient through minimization problems. As a consequence, we obtain methods able to tackle both mean-field games and mean-field control problems in moderate dimension. We analyze the numerical behavior of our algorithms on several multidimensional examples including non linear quadratic models. |
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| AbstractList | We propose several algorithms to solve McKean-Vlasov Forward Backward Stochastic Differential Equations (FBSDEs). Our schemes rely on the approximating power of neural networks to estimate the solution or its gradient through minimization problems. As a consequence, we obtain methods able to tackle both mean-field games and mean-field control problems in moderate dimension. We analyze the numerical behavior of our algorithms on several examples including non linear quadratic models. We propose several algorithms to solve McKean-Vlasov Forward Backward Stochastic Differential Equations (FBSDEs). Our schemes rely on the approximating power of neural networks to estimate the solution or its gradient through minimization problems. As a consequence, we obtain methods able to tackle both mean-field games and mean-field control problems in moderate dimension. We analyze the numerical behavior of our algorithms on several multidimensional examples including non linear quadratic models. |
| Author | Germain, Maximilien Warin, Xavier Mikael, Joseph |
| Author_xml | – sequence: 1 givenname: Maximilien orcidid: 0000-0003-3231-2087 surname: Germain fullname: Germain, Maximilien email: mgermain@lpsm.paris organization: EDF R&D, Université de Paris, LPSM – sequence: 2 givenname: Joseph surname: Mikael fullname: Mikael, Joseph organization: EDF R&D – sequence: 3 givenname: Xavier surname: Warin fullname: Warin, Xavier organization: EDF R&D, FiME |
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| Cites_doi | 10.1007/s11579-017-0206-z 10.1137/090758477 10.1051/proc/201965084 10.1016/j.crma.2006.09.019 10.1016/j.crma.2006.09.018 10.1186/s41546-020-00047-w 10.1007/s10915-019-00908-3 10.1214/18-AAP1429 10.1137/20M1316640 10.1073/pnas.1718942115 10.1214/14-AAP1020 10.1090/psapm/078/06 10.3389/fams.2020.00011 10.1007/978-3-319-56436-4 10.1007/s42985-020-00062-8 10.1109/MIS.2020.2971597 10.1090/mcom/3514 10.1007/s00332-018-9525-3 10.1016/j.spa.2004.01.001 10.1007/s11009-019-09767-9 10.1137/120883499 10.1214/105051605000000412 10.1007/978-3-319-58920-6 |
| ContentType | Journal Article |
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| Keywords | MSC 68T07 Mean-field games Neural networks 49N80 MSC 35Q89 Machine learning MSC 65C30 Deep BSDE McKean-Vlasov FBSDEs machine learning mean-field games |
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| References_xml | – reference: Han J, Jentzen A, Weinan E (2017) Solving high-dimensional partial differential equations using deep learning. Proc Nat Acad Sci 115. https://doi.org/10.1073/pnas.1718942115 – reference: HuréCPhamHWarinXDeep backward schemes for high-dimensional nonlinear PDEsMath Comput20208932415471579408191110.1090/mcom/35141440.60063 – reference: Kingma D, Ba J (2015) Adam: A method for stochastic optimization. International Conference on Learning Representations – reference: Lasry JM, Lions PL (2006b) Jeux à champ moyen. ii – horizon fini et contrôle optimal. Comptes Rendus Mathématique Académie des Sciences, Paris 10. https://doi.org/10.1016/j.crma.2006.09.018 – reference: Beck C, Becker S, Cheridito P, Jentzen A, Neufeld A (2019a) Deep splitting method for parabolic PDEs. arXiv preprint: arXiv:190703452 – reference: Sergeev A, Del Balso M (2018) Horovod: Fast and easy distributed deep learning in tensorflow – reference: Achdou Y, Kobeissi Z (2020) Mean field games of controls: Finite difference approximations. arXiv:200303968 – reference: Pham H, Warin X, Germain M (2021) Neural networks-based backward scheme for fully nonlinear PDEs. SN Part Diff Equations Appl 2. https://doi.org/10.1007/s42985-020-00062-8 – reference: Carmona R, Laurière M (2019) Convergence analysis of machine learning algorithms for the numerical solution of mean field control and games: II – the finite horizon case. arXiv preprint arXiv:190801613, to appear in The Annals of Applied Probability – reference: Fouque JP, Zhang Z (2020) Deep learning methods for mean field control problems with delay. Front Appl Math Stat 6. https://doi.org/10.3389/fams.2020.00011 – reference: Angiuli A, Graves CV, Li H, Chassagneux JF, Delarue F, Carmona R (2019) Cemracs 2017: Numerical probabilistic approach to MFG. ESAIM: Proc Surv 65:84–113. https://doi.org/10.1051/proc/201965084 – reference: Carmona R, Lacker D (2015) A probabilistic weak formulation of mean field games and applications. Ann Appl Prob 25(3):1189–1231. https://doi.org/10.1214/14-AAP1020 – reference: HanJLongJConvergence of the deep BSDE method for coupled FBSDEsProb Uncert Quan Risk202051133412222710.1186/s41546-020-00047-w1454.60105 – reference: BachouchAHuréCPhamHLangrenéNDeep neural networks algorithms for stochastic control problems on finite horizon: Numerical computationsMethodol Comput Appl Probab202110.1007/s11009-019-09767-91466.65007 – reference: Beck C, Jentzen A (2019b) Machine learning approximation algorithms for high-dimensional fully nonlinear partial differential equations and second-order backward stochastic differential equations. J Nonlinear Sci 29(4):1563–1619. https://doi.org/10.1007/s00332-018-9525-3 – reference: BouchardBTouziNDiscrete-time approximation and monte-carlo simulation of backward stochastic differential equationsStochastic Process Appl20041112175206205653610.1016/j.spa.2004.01.0011071.60059 – reference: Huré C, Pham H, Bachouch A, Langrené N (2021) Deep neural networks algorithms for stochastic control problems on finite horizon: convergence analysis. SIAM J Numer Anal 59(1):525–557. https://doi.org/10.1137/20M1316640 – reference: Carmona R, Delarue F (2018a) Probabilistic theory of mean field games with applications I. Springer. https://doi.org/10.1007/978-3-319-58920-6 – reference: Carmona R, Delarue F (2013) Probabilistic analysis of mean-field games. 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| Title | Numerical Resolution of McKean-Vlasov FBSDEs Using Neural Networks |
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