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

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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

Abstract
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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 & Continuous Dynamical Systems - A, 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. 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. doi: 10.1137/120882421.
[3]	Y. Achdou and I. C. Dolcetta, Mean field games: Numerical methods,, SIAM Journal of Numerical Analysis, 48 (2010), 1136. 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, (2008).
[5]	G. Barles and P. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations,, Asymptotic Anal., 4 (1991), 271.
[6]	B. Bouchard and N. Touzi, Weak dynamic programming principle for viscosity solutions,, SIAM Journal on Control and Optimization, 49 (2011), 948. doi: 10.1137/090752328.
[7]	L. Breiman, Probability,, Addison-Wesley Publishing Company, (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.
[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. 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), 3115. 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. doi: 10.1137/120902987.
[13]	P. G. Ciarlet and J.-L. Lions (eds.), Handbook of Numerical Analysis. Vol. II,, Handbook of Numerical Analysis, (1991).
[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. 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. 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. 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. doi: 10.1016/j.jfa.2007.09.020.
[19]	W. Fleming and H. Soner, Controlled Markov Processes and Viscosity Solutions,, Springer, (1993).
[20]	D. Gomes and J. Saúde, Mean field models, a brief survey,, Dynamic Games and Applications, 4 (2014), 110. 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). 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, (2011), 205. 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. 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, (2001). 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. 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. 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. 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. doi: 10.1007/s11537-007-0657-8.
[30]	A. Quarteroni, R. Sacco and F. Saleri, Numerical Mathematics (Second Ed.),, Springer, (2007).
[31]	C. Villani, Topics in Optimal Transportation,, Vol. 58 of Graduate Studies in Mathematics. American Mathematical Society, (2003).
[32]	J. Yong and X. Zhou, Stochastic Controls, Hamiltonian Systems and HJB Equations,, Springer-Verlag, (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. 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. doi: 10.1137/120882421.
[3]	Y. Achdou and I. C. Dolcetta, Mean field games: Numerical methods,, SIAM Journal of Numerical Analysis, 48 (2010), 1136. 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, (2008).
[5]	G. Barles and P. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations,, Asymptotic Anal., 4 (1991), 271.
[6]	B. Bouchard and N. Touzi, Weak dynamic programming principle for viscosity solutions,, SIAM Journal on Control and Optimization, 49 (2011), 948. doi: 10.1137/090752328.
[7]	L. Breiman, Probability,, Addison-Wesley Publishing Company, (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.
[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. 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), 3115. 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. doi: 10.1137/120902987.
[13]	P. G. Ciarlet and J.-L. Lions (eds.), Handbook of Numerical Analysis. Vol. II,, Handbook of Numerical Analysis, (1991).
[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. 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. 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. 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. doi: 10.1016/j.jfa.2007.09.020.
[19]	W. Fleming and H. Soner, Controlled Markov Processes and Viscosity Solutions,, Springer, (1993).
[20]	D. Gomes and J. Saúde, Mean field models, a brief survey,, Dynamic Games and Applications, 4 (2014), 110. 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). 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, (2011), 205. 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. 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, (2001). 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. 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. 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. 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. doi: 10.1007/s11537-007-0657-8.
[30]	A. Quarteroni, R. Sacco and F. Saleri, Numerical Mathematics (Second Ed.),, Springer, (2007).
[31]	C. Villani, Topics in Optimal Transportation,, Vol. 58 of Graduate Studies in Mathematics. American Mathematical Society, (2003).
[32]	J. Yong and X. Zhou, Stochastic Controls, Hamiltonian Systems and HJB Equations,, Springer-Verlag, (1999). doi: 10.1007/978-1-4612-1466-3.

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