September  2015, 35(9): 4269-4292. doi: 10.3934/dcds.2015.35.4269

A semi-Lagrangian scheme for a degenerate second order mean field game system

1. 

"Sapienza" Università di Roma, Dipartimento di Matematica, P.le A. Moro 5, 00185 Roma

2. 

XLIM - DMI UMR CNRS 7252, Faculté des Sciences et Techniques, Université de Limoges, 123 Avenue Albert Thomas, 87060-Limoges Cedes, France

Received  April 2014 Revised  September 2014 Published  April 2015

In this paper we study a fully discrete Semi-Lagrangian approximation of a second order Mean Field Game system, which can be degenerate. We prove that the resulting scheme is well posed and, if the state dimension is equals to one, we prove a convergence result. Some numerical simulations are provided, evidencing the convergence of the approximation and also the difference between the numerical results for the degenerate and non-degenerate cases.
Citation: Elisabetta Carlini, Francisco J. Silva. A semi-Lagrangian scheme for a degenerate second order mean field game system. Discrete & Continuous Dynamical Systems, 2015, 35 (9) : 4269-4292. doi: 10.3934/dcds.2015.35.4269
References:
[1]

Y. Achdou, F. Camilli and L. Corrias, On numerical approximations of the Hamilton-Jacobi-transport system arising in high frequency, Discrete and Continuous Dynamical Systems- Series B, 19 (2014), 629-650. doi: 10.3934/dcdsb.2014.19.629.  Google Scholar

[2]

Y. Achdou, F. Camilli and I. C. Dolcetta, Mean field games: Convergence of a finite difference method, SIAM J. Numer. Anal., 51 (2013), 2585-2612. doi: 10.1137/120882421.  Google Scholar

[3]

Y. Achdou and I. C. Dolcetta, Mean field games: Numerical methods, SIAM Journal of Numerical Analysis, 48 (2010), 1136-1162. doi: 10.1137/090758477.  Google Scholar

[4]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Second edition. Lecture notes in Mathematics ETH Zürich. Birkhäuser Verlag, Bassel, 2008.  Google Scholar

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G. Barles and P. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations, Asymptotic Anal., 4 (1991), 271-283.  Google Scholar

[6]

B. Bouchard and N. Touzi, Weak dynamic programming principle for viscosity solutions, SIAM Journal on Control and Optimization, 49 (2011), 948-962. doi: 10.1137/090752328.  Google Scholar

[7]

L. Breiman, Probability, Addison-Wesley Publishing Company, Reading, MA, 1968.  Google Scholar

[8]

F. Camilli and M. Falcone, An approximation scheme for the optimal control of diffusion processes, RAIRO Modél. Math. Anal. Numér., 29 (1995), 97-122.  Google Scholar

[9]

F. Camilli and F. J. Silva, A semi-discrete in time approximation for a first order-finite mean field game problem, Network and Heterogeneous Media, 7 (2012), 263-277. doi: 10.3934/nhm.2012.7.263.  Google Scholar

[10]

P. Cardaliaguet, Notes on Mean Field Games: From P.-L. Lions' lectures at Collège de France,, Lecture Notes given at Tor Vergata., ().   Google Scholar

[11]

E. Carlini and F. J. Silva, Semi-lagrangian schemes for mean field game models, in Decision and Control (CDC), 2013 IEEE 52nd Annual Conference on, (2013), 3115-3120. doi: 10.1109/CDC.2013.6760358.  Google Scholar

[12]

E. Carlini and F. J. Silva, A fully discrete semi-lagrangian scheme for a first order mean field game problem, SIAM Journal on Numerical Analysis, 52 (2014), 45-67. doi: 10.1137/120902987.  Google Scholar

[13]

P. G. Ciarlet and J.-L. Lions (eds.), Handbook of Numerical Analysis. Vol. II, Handbook of Numerical Analysis, II, North-Holland, Amsterdam, 1991, Finite element methods. Part 1.  Google Scholar

[14]

M. Crandall, H. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. American Mathematical Society (New Series), 27 (1992), 1-67. doi: 10.1090/S0273-0979-1992-00266-5.  Google Scholar

[15]

F. Da Lio and O. Ley, Uniqueness results for second-order bellman-isaacs equations under quadratic growth assumptions and applications, SIAM Journal on Control and Optimization, 45 (2006), 74-106. doi: 10.1137/S0363012904440897.  Google Scholar

[16]

K. Debrabant and E. R. Jakobsen, Semi-Lagrangian schemes for linear and fully non-linear diffusion equations, Math. Comp., 82 (2013), 1433-1462. doi: 10.1090/S0025-5718-2012-02632-9.  Google Scholar

[17]

M. Falcone and R. Ferretti, Semi-Lagrangian Approximation Schemes for Linear and Hamilton-Jacobi Equations, MOS-SIAM Series on Optimization, 2013. doi: 10.1137/1.9781611973051.  Google Scholar

[18]

A. Figalli, Existence and uniqueness of martingale solutions for sdes with rough or degenerate coefficients, J. Funct. Anal., 254 (2008), 109-153. doi: 10.1016/j.jfa.2007.09.020.  Google Scholar

[19]

W. Fleming and H. Soner, Controlled Markov Processes and Viscosity Solutions, Springer, New York, 1993.  Google Scholar

[20]

D. Gomes and J. Saúde, Mean field models, a brief survey, Dynamic Games and Applications, 4 (2014), 110-154. doi: 10.1007/s13235-013-0099-2.  Google Scholar

[21]

O. Guéant, Mean field games equations with quadratic hamiltonian: A specific approach, Mathematical Models and Methods in Applied Sciences, 22 (2012), 1250022, 37pp. doi: 10.1142/S0218202512500224.  Google Scholar

[22]

O. Guéant, J.-M. Lasry and P.-L. Lions, Mean field games and applications, in Paris-Princeton Lectures on Mathematical Finance 2010, vol. 2003 of Lecture Notes in Math., Springer, Berlin, (2011), 205-266. doi: 10.1007/978-3-642-14660-2_3.  Google Scholar

[23]

M. Huang, P. Caines and R. Malhamé, Individual and mass behavior in large population stochastic wireless power control problems: Centralized and Nash equillibrium solutions,, Proc. 42nd IEEE-CDC., ().   Google Scholar

[24]

M. Huang, P. Caines and R. Malhamé, Large populations stochastic dynamics games: Closed-loop McKean-Vlasov systems and the Nash certainly equivalence principle, Comm. Inf. Syst., 6 (2006), 221-251. doi: 10.4310/CIS.2006.v6.n3.a5.  Google Scholar

[25]

H. Kushner and P. Dupuis, Numerical Methods for Stochastic Control Problems in Continuous Time, vol. 24 of Applications of mathematics, Springer, New York, 2001, Second edition. doi: 10.1007/978-1-4613-0007-6.  Google Scholar

[26]

A. Lachapelle, J. Salomon and G. Turinici, Computation of mean field equilibria in economics, Mathematical Models and Methods in Applied Sciences, 20 (2010), 567-588. doi: 10.1142/S0218202510004349.  Google Scholar

[27]

J.-M. Lasry and P.-L. Lions, Jeux à champ moyen I. Le cas stationnaire, C. R. Math. Acad. Sci. Paris, 343 (2006), 619-625. doi: 10.1016/j.crma.2006.09.019.  Google Scholar

[28]

J.-M. Lasry and P.-L. Lions, Jeux à champ moyen II. Horizon fini et contrôle optimal, C. R. Math. Acad. Sci. Paris, 343 (2006), 679-684. doi: 10.1016/j.crma.2006.09.018.  Google Scholar

[29]

J.-M. Lasry and P.-L. Lions, Mean field games, Jpn. J. Math., 2 (2007), 229-260. doi: 10.1007/s11537-007-0657-8.  Google Scholar

[30]

A. Quarteroni, R. Sacco and F. Saleri, Numerical Mathematics (Second Ed.), Springer, Berlin, 2007.  Google Scholar

[31]

C. Villani, Topics in Optimal Transportation, Vol. 58 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2003.  Google Scholar

[32]

J. Yong and X. Zhou, Stochastic Controls, Hamiltonian Systems and HJB Equations, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-1466-3.  Google Scholar

show all references

References:
[1]

Y. Achdou, F. Camilli and L. Corrias, On numerical approximations of the Hamilton-Jacobi-transport system arising in high frequency, Discrete and Continuous Dynamical Systems- Series B, 19 (2014), 629-650. doi: 10.3934/dcdsb.2014.19.629.  Google Scholar

[2]

Y. Achdou, F. Camilli and I. C. Dolcetta, Mean field games: Convergence of a finite difference method, SIAM J. Numer. Anal., 51 (2013), 2585-2612. doi: 10.1137/120882421.  Google Scholar

[3]

Y. Achdou and I. C. Dolcetta, Mean field games: Numerical methods, SIAM Journal of Numerical Analysis, 48 (2010), 1136-1162. doi: 10.1137/090758477.  Google Scholar

[4]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Second edition. Lecture notes in Mathematics ETH Zürich. Birkhäuser Verlag, Bassel, 2008.  Google Scholar

[5]

G. Barles and P. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations, Asymptotic Anal., 4 (1991), 271-283.  Google Scholar

[6]

B. Bouchard and N. Touzi, Weak dynamic programming principle for viscosity solutions, SIAM Journal on Control and Optimization, 49 (2011), 948-962. doi: 10.1137/090752328.  Google Scholar

[7]

L. Breiman, Probability, Addison-Wesley Publishing Company, Reading, MA, 1968.  Google Scholar

[8]

F. Camilli and M. Falcone, An approximation scheme for the optimal control of diffusion processes, RAIRO Modél. Math. Anal. Numér., 29 (1995), 97-122.  Google Scholar

[9]

F. Camilli and F. J. Silva, A semi-discrete in time approximation for a first order-finite mean field game problem, Network and Heterogeneous Media, 7 (2012), 263-277. doi: 10.3934/nhm.2012.7.263.  Google Scholar

[10]

P. Cardaliaguet, Notes on Mean Field Games: From P.-L. Lions' lectures at Collège de France,, Lecture Notes given at Tor Vergata., ().   Google Scholar

[11]

E. Carlini and F. J. Silva, Semi-lagrangian schemes for mean field game models, in Decision and Control (CDC), 2013 IEEE 52nd Annual Conference on, (2013), 3115-3120. doi: 10.1109/CDC.2013.6760358.  Google Scholar

[12]

E. Carlini and F. J. Silva, A fully discrete semi-lagrangian scheme for a first order mean field game problem, SIAM Journal on Numerical Analysis, 52 (2014), 45-67. doi: 10.1137/120902987.  Google Scholar

[13]

P. G. Ciarlet and J.-L. Lions (eds.), Handbook of Numerical Analysis. Vol. II, Handbook of Numerical Analysis, II, North-Holland, Amsterdam, 1991, Finite element methods. Part 1.  Google Scholar

[14]

M. Crandall, H. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. American Mathematical Society (New Series), 27 (1992), 1-67. doi: 10.1090/S0273-0979-1992-00266-5.  Google Scholar

[15]

F. Da Lio and O. Ley, Uniqueness results for second-order bellman-isaacs equations under quadratic growth assumptions and applications, SIAM Journal on Control and Optimization, 45 (2006), 74-106. doi: 10.1137/S0363012904440897.  Google Scholar

[16]

K. Debrabant and E. R. Jakobsen, Semi-Lagrangian schemes for linear and fully non-linear diffusion equations, Math. Comp., 82 (2013), 1433-1462. doi: 10.1090/S0025-5718-2012-02632-9.  Google Scholar

[17]

M. Falcone and R. Ferretti, Semi-Lagrangian Approximation Schemes for Linear and Hamilton-Jacobi Equations, MOS-SIAM Series on Optimization, 2013. doi: 10.1137/1.9781611973051.  Google Scholar

[18]

A. Figalli, Existence and uniqueness of martingale solutions for sdes with rough or degenerate coefficients, J. Funct. Anal., 254 (2008), 109-153. doi: 10.1016/j.jfa.2007.09.020.  Google Scholar

[19]

W. Fleming and H. Soner, Controlled Markov Processes and Viscosity Solutions, Springer, New York, 1993.  Google Scholar

[20]

D. Gomes and J. Saúde, Mean field models, a brief survey, Dynamic Games and Applications, 4 (2014), 110-154. doi: 10.1007/s13235-013-0099-2.  Google Scholar

[21]

O. Guéant, Mean field games equations with quadratic hamiltonian: A specific approach, Mathematical Models and Methods in Applied Sciences, 22 (2012), 1250022, 37pp. doi: 10.1142/S0218202512500224.  Google Scholar

[22]

O. Guéant, J.-M. Lasry and P.-L. Lions, Mean field games and applications, in Paris-Princeton Lectures on Mathematical Finance 2010, vol. 2003 of Lecture Notes in Math., Springer, Berlin, (2011), 205-266. doi: 10.1007/978-3-642-14660-2_3.  Google Scholar

[23]

M. Huang, P. Caines and R. Malhamé, Individual and mass behavior in large population stochastic wireless power control problems: Centralized and Nash equillibrium solutions,, Proc. 42nd IEEE-CDC., ().   Google Scholar

[24]

M. Huang, P. Caines and R. Malhamé, Large populations stochastic dynamics games: Closed-loop McKean-Vlasov systems and the Nash certainly equivalence principle, Comm. Inf. Syst., 6 (2006), 221-251. doi: 10.4310/CIS.2006.v6.n3.a5.  Google Scholar

[25]

H. Kushner and P. Dupuis, Numerical Methods for Stochastic Control Problems in Continuous Time, vol. 24 of Applications of mathematics, Springer, New York, 2001, Second edition. doi: 10.1007/978-1-4613-0007-6.  Google Scholar

[26]

A. Lachapelle, J. Salomon and G. Turinici, Computation of mean field equilibria in economics, Mathematical Models and Methods in Applied Sciences, 20 (2010), 567-588. doi: 10.1142/S0218202510004349.  Google Scholar

[27]

J.-M. Lasry and P.-L. Lions, Jeux à champ moyen I. Le cas stationnaire, C. R. Math. Acad. Sci. Paris, 343 (2006), 619-625. doi: 10.1016/j.crma.2006.09.019.  Google Scholar

[28]

J.-M. Lasry and P.-L. Lions, Jeux à champ moyen II. Horizon fini et contrôle optimal, C. R. Math. Acad. Sci. Paris, 343 (2006), 679-684. doi: 10.1016/j.crma.2006.09.018.  Google Scholar

[29]

J.-M. Lasry and P.-L. Lions, Mean field games, Jpn. J. Math., 2 (2007), 229-260. doi: 10.1007/s11537-007-0657-8.  Google Scholar

[30]

A. Quarteroni, R. Sacco and F. Saleri, Numerical Mathematics (Second Ed.), Springer, Berlin, 2007.  Google Scholar

[31]

C. Villani, Topics in Optimal Transportation, Vol. 58 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2003.  Google Scholar

[32]

J. Yong and X. Zhou, Stochastic Controls, Hamiltonian Systems and HJB Equations, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-1466-3.  Google Scholar

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