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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

Elisabetta Carlini 1, and  Francisco J. Silva 2,

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.
Keywords: degenerate second order system, Mean field games, numerical methods., semi-Lagrangian schemes, convergence analysis.
Mathematics Subject Classification: Primary: 65M12, 91A13; Secondary: 65M25, 91A23, 49J15, 35F21, 35Q8.
Citation: Elisabetta Carlini, Francisco J. Silva. A semi-Lagrangian scheme for a degenerate second order mean field game system. Discrete and 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.

[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.

[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.

[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.

[5]

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

[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.

[7]

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

[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.

[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.

[10]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[19]

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

[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.

[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.

[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.

[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., (). 

[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.

[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.

[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.

[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.

[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.

[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.

[30]

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

[31]

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

[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.

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.

[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.

[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.

[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.

[5]

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

[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.

[7]

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

[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.

[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.

[10]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[19]

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

[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.

[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.

[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.

[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., (). 

[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.

[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.

[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.

[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.

[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.

[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.

[30]

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

[31]

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

[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.

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