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Approximation of $ N $-player stochastic games with singular controls by mean field games

  • *Corresponding author: Xin Guo

    *Corresponding author: Xin Guo 
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  • This paper establishes that a class of $ N $-player stochastic games with singular controls, either of bounded velocity or of finite variation, can both be approximated by mean field games (MFGs) with singular controls of bounded velocity. More specifically, it shows (i) the optimal control to an MFG with singular controls of a bounded velocity $ \theta $ is shown to be an $ \epsilon_N $-NE to an $ N $-player game with singular controls of the bounded velocity, with $ \epsilon_N = O(\frac{1}{\sqrt{N}}) $, and (ii) the optimal control to this MFG is an $ (\epsilon_N + \epsilon_{\theta}) $-NE to an $ N $-player game with singular controls of finite variation, where $ \epsilon_{\theta} $ is an error term that depends on $ \theta $. This work generalizes the classical result on approximation $ N $-player games by MFGs, by allowing for discontinuous controls.

    Mathematics Subject Classification: Primary: 49N80, 91A15; Secondary: 91A06, 91A16.


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  • [1] E. Bayraktar and A. Cohen, Analysis of a finite state many player game using its master equation, SIAM Journal on Control and Optimization, 56 (2018), 3538-3568.  doi: 10.1137/17M113887X.
    [2] E. BayraktarA. Budhiraja and A. Cohen, A numerical scheme for a mean field game in some queueing systems based on Markov chain approximation method, SIAM Journal on Control and Optimization, 56 (2018), 4017-4044.  doi: 10.1137/17M1154357.
    [3] A. Bensoussan, J. F. Jens and P. Yam, Mean Field Games and Mean Field Type Control Theory, Springer, 2013. doi: 10.1007/978-1-4614-8508-7.
    [4] P. Billingsley, Convergence of Probability Measures, John Wiley & Sons, 2013.
    [5] P. Cardaliaguet, Notes on mean field games (from Pierre-Louis Lions' lectures at College de France), Technical Report, 2013.
    [6] P. Cardaliaguet, The convergence problem in mean field games with local coupling, Applied Mathematics $ & $ Optimization, 76 (2017), 177-215.  doi: 10.1007/s00245-017-9434-0.
    [7] P. Cardaliaguet, F. Delarue, J-M. Lasry and P-L, Lions, The Master Equation and the Convergence Problem in Mean Field Games:(AMS-201), Vol. 201, Princeton University Press, 2019. doi: 10.2307/j.ctvckq7qf.
    [8] R. Carmona and F. Delarue, Probabilistic Theory of Mean Field Games with Applications Ⅰ-Ⅱ, Springer, 2018.
    [9] A. Cecchin and G. Pelino, Convergence, fluctuations and large deviations for finite state mean field games via the master equation, Stochastic Processes and their Applications, 129 (2019), 4510-4555.  doi: 10.1016/j.spa.2018.12.002.
    [10] A. CecchinP. D. PraM. Fischer and G. Pelino, On the convergence problem in mean field games: a two state model without uniqueness, SIAM Journal on Control and Optimization, 57 (2019), 2443-2466.  doi: 10.1137/18M1222454.
    [11] F. DelarueD. Lacker and K. Ramanan, From the master equation to mean field game limit theory: a central limit theorem, Electronic Journal of Probability, 24 (2019), 1-54.  doi: 10.1214/19-EJP298.
    [12] F. DelarueD. Lacker and K. Ramanan, From the master equation to mean field game limit theory: large deviations and concentration of measure, The Annals of Probability, 48 (2020), 211-263.  doi: 10.1214/19-AOP1359.
    [13] J. Dianetti and G. Ferrari, Nonzero-sum submodular monotone-follower games: existence and approximation of Nash equilibria, SIAM Journal on Control and Optimization, 58 (2020), 1257-1288.  doi: 10.1137/19M1238782.
    [14] N. El KarouiC. KapoudjianE. PardouxS. 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.
    [15] W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control, Vol. 1, Springer Science $&$ Business Media, 2012.
    [16] G. X. Fu and U. Horst, Mean field games with singular controls, SIAM Journal on Control and Optimization, 55 (2017), 3833-3868.  doi: 10.1137/17M1123742.
    [17] O. Guéant, J. Lasry and P. L. Lions, Mean field games and applications, in Paris-Princeton Lectures on Mathematical Finance 2010, Springer, Berlin, Heidelberg, (2011), 205-266. doi: 10.1007/978-3-642-14660-2_3.
    [18] X. Guo and R. Xu, Stochastic games for fuel followers problem: N vs MFG, SIAM Journal of Control and Optimization, 57 (2019), 659-692.  doi: 10.1137/17M1159531.
    [19] D. Hernández-HernándezJ. L. Pérez and K. Yamazaki, Optimality of refraction strategies for spectrally negative Lévy processes, SIAM Journal on Control and Optimization, 54 (2016), 1126-1156.  doi: 10.1137/15M1051208.
    [20] M. HuangR. P. Malhamé and P. E. Caines, Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle, Communications in Information $ &$ Systems, 6 (2006), 221-252. 
    [21] D. Lacker and T. Zariphopoulou, Mean field and N-agent games for optimal investment under relative performance criteria, Mathematical Finance, 29 (2019), 1003-1038.  doi: 10.1111/mafi.12206.
    [22] D. Lacker, Mean field games via controlled martingale problems: existence of Markovian equilibria, Stochastic Processes and their Applications, 125 (2015), 2856-2894.  doi: 10.1016/j.spa.2015.02.006.
    [23] D. Lakcer, A general characterization of the mean field limit for stochastic differential games, Probability Theory and Related Fields, 165 (2016), 581-648.  doi: 10.1007/s00440-015-0641-9.
    [24] D. Lakcer, On the convergence of closed-loop Nash equilibria to the mean field game limit, The Annals of Applied Probability, 30 (2020), 1693-1761.  doi: 10.1214/19-AAP1541.
    [25] J. Lasry and P. L. Lions, Mean field games, Japanese Journal of Mathematics, 2 (2007), 229-260.  doi: 10.1007/s11537-007-0657-8.
    [26] M. Laurière and L. Tangpi, Convergence of large population games to mean field games with interaction through the controls, SIAM Journal on Mathematical Analysis, 54 (2022), 3535-3574.  doi: 10.1137/22M1469328.
    [27] J. L. Menaldi and M. I. Taksar, Optimal correction problem of a multidimensional stochastics system, Automatica, 25 (1989), 223-232.  doi: 10.1016/0005-1098(89)90075-7.
    [28] M. NutzJ. S. Martin and X. Tan, Convergence to the mean field game limit: a case study, The Annals of Applied Probability, 30 (2020), 259-286.  doi: 10.1214/19-AAP1501.
    [29] H. Pham, Continuous-Time Stochastic Control and Optimization with Financial Applications, Springer, 2009. doi: 10.1007/978-3-540-89500-8.
    [30] D. R. Smart, Fixed Point Theorems, Vol. 66, CUP Archive, 1980.
    [31] A. K. Zvonkin, A transformation of the phase space of a diffusion process that removes the drift, Mathematics of the USSR-Sbornik, 22 (1974), 129. 
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