Abstract
The purpose of this note is to show that a common noise may restore uniqueness in mean field games. To this end, we focus on a class of examples driven by linear dynamics and quadratic cost functions. Given these linear-quadratic mean field games, we prove existence and uniqueness of solutions in the presence of common noise and construct a counter-example in the absence of common noise. This illustrates the principle, already observed in dynamical systems like ODEs, that introducing an appropriate noise may restore uniqueness.
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References
Cardaliaguet P (2013) Note on mean-field games. Technical report. https://www.ceremade.dauphine.fr/~cardalia/MFG20130420.pdf
Carmona R, Delarue F (2013) Probabilistic analysis of mean-field games. SIAM J Control Optim 51(4):2705–2734
Carmona R, Delarue F, Lachapelle A (2013) Control of McKean–Vlasov dynamics versus mean field games. Math Financ Econ 7(2):131–166
Delarue F (2002) On the existence and uniqueness of solutions to FBSDEs in a non-degenerate case. Stoch Proc Appl 99(2):209–286. doi:10.1016/S0304-4149(02)00085-6
Flandoli F (2011) Random perturbation of PDEs and fluid dynamic models, volume 2015 of Lecture notes in mathematics. Springer, Heidelberg. Lectures from the 40th Probability Summer School held in Saint-Flour 2010
Huang M (2010) Large-population lqg games involving a major player: the nash certainty equivalence principle. SIAM J Control Optim 48(5):3318–3353. doi:10.1137/080735370
Huang M, Malhamé RP, Caines PE (2006) Large population stochastic dynamic games: closed-loop McKean–Vlasov systems and the Nash certainty equivalence principle. Commun Inf Syst 6(3):221–251
Lasry J-M, Lions P-L (2007) Mean field games. Jpn J Math 2(1):229–260
Ma J, Protter P, Yong JM (1994) Solving forward-backward stochastic differential equations explicitly—a four step scheme. Probab Theory Related Fields 98(3):339–359
Nguyen SL, Huang M (2011) Mean field lqg games with a major player: continuum parameters for minor players. In: 50th IEEE conference on decision and control and European control conference, pp 1012–1017. doi:10.1109/CDC.2011.6160790
Pham H (2009) Continuous-time stochastic control and optimization with financial applications, volume 61 of Stochastic modelling and applied probability. Springer, Berlin
Sznitman A-S (1991) Topics in propagation of chaos. In: Hennequin P-L (ed) Ecole d’Eté de Probabilités de Saint-Flour XIX–1989. Lecture Notes in Mathematics, vol 1464, pp 165–251
Yong J (1999) Linear forward-backward stochastic differential equations. Appl Math Optim 39(1):93–119
Yong J (2006) Linear forward-backward stochastic differential equations with random coefficients. Probab Theory Related Fields 135(1):53–83
Yong J, Zhou XY (1999) Stochastic controls, volume 43 of Applications of mathematics. Springer, New York. Hamiltonian systems and HJB equations
Zhu X, Carmona R (2014) A probabilistic approach to mean field games with major and minor players
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Tchuendom, R.F. Uniqueness for Linear-Quadratic Mean Field Games with Common Noise. Dyn Games Appl 8, 199–210 (2018). https://doi.org/10.1007/s13235-016-0200-8
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DOI: https://doi.org/10.1007/s13235-016-0200-8