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On the near-viability property of controlled mean-field flows

  • *Corresponding author: Juan Li

    *Corresponding author: Juan Li
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  • We aim at studying the property of controlled stochastic flows with mean-field dynamics to comply with some (closed) state restrictions. This property, known as (near)-viability, is tackled via (quasi-)tangency methods. Law restrictions and mixed state-law restrictions are considered as the interplay between the two classes. As an auxiliary result used in this process, Theorem 1.2, whose importance exceeds the present framework, dissociates, through a class of elementary controls, the contribution of the Brownian filtration and that of the initial $ \sigma $-field. Explicit conditions for the coefficient functions are provided in the invariance context. Moreover, specific applications to comparison in the convex order illustrate the theoretical results.

    Mathematics Subject Classification: 93E20; 60H10; 60K35.

    Citation:

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  • [1] J. P. Aubin and G. Da Prato, Stochastic viability and invariance, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 17 (1990), 595-613. 
    [2] J. P. Aubin and H. Frankowska, Set-Valued Analysis, Volume 2 of Systems & Control: Foundations & Applications. Birkhäuser Boston Inc., Boston, MA, 1990.
    [3] M. Bossy and D. Talay, A stochastic particle method for the McKean-Vlasov and the Burgers equation, Math. Comput., 66 (1997), 157-192.  doi: 10.1090/S0025-5718-97-00776-X.
    [4] R. BuckdahnP. Cardaliaguet and M. Quincampoix, A representation formula for the mean curvature motion, SIAM J. Math. Anal., 33 (2001), 827-846.  doi: 10.1137/S0036141000380334.
    [5] R. BuckdahnY. Chen and J. Li, Partial derivative with respect to the measure and its application to general controlled mean-field systems, Stochastic Process. Appl., 134 (2021), 265-307.  doi: 10.1016/j.spa.2021.01.003.
    [6] R. BuckdahnS. PengM. Quincampoix and C. Rainer, Existence of stochastic control under state constraints, C. R. Acad. Sci. Paris, Sér. I Math, 327 (1998), 17-22.  doi: 10.1016/S0764-4442(98)80096-7.
    [7] R. BuckdahnM. Quincampoix and G. Tessitore, Controlled stochastic differential equations under constraints in infinite dimensional spaces, SIAM J. Control Optim., 47 (2008), 218-250.  doi: 10.1137/060674284.
    [8] R. BuckdahnJ. Li and S. Peng, Mean-field backward stochastic differential equations and related partial differential equations, Stochastic Process. Appl., 119 (2009), 3133-3154.  doi: 10.1016/j.spa.2009.05.002.
    [9] R. BuckdahnJ. LiS. Peng and C. Rainer, Mean-field stochastic differential equations and associated PDEs, Ann. Probab., 45 (2017), 824-878.  doi: 10.1214/15-AOP1076.
    [10] P. Cardaliaguet, Notes on mean field games, P. L. Lions' Lectures at Collège de France, 2013.
    [11] O. Carja, M. Necula and I. I. Vrabie, Viability, invariance and applications, volume 207 of North-Holl. Math. Stud., North-Holland, 2007.
    [12] O. Cârjǎ and Ioan I. Vrabie, Some new viability results for semilinear differential inclusions, NODEA-Nonlinear Diff., 4 (1997), 401-424.  doi: 10.1007/s000300050022.
    [13] T. Chan, Dynamics of the Mckean-Vlasov equation, Ann. Probab., 22 (1994), 431-441. 
    [14] J. F. Chassagneux, D. Crisan and F. Delarue, A Probabilistic Approach to Classical Solutions of the Master Equation for Large Population Equilibria, Memoirs of the American Mathematical Society, 2022. doi: 10.1090/memo/1379.
    [15] D. A. Dawsont and J. Gärtner, Large deviations from the McKean-Vlasov limit for weakly interacting diffusions, Stochastics, 20 (1987), 247-308.  doi: 10.1080/17442508708833446.
    [16] S. Ditlevsen and E. Löcherbach, Multi-class oscillating systems of interacting neurons, Stochastic Process. Appl., 127 (2017), 1840-1869.  doi: 10.1016/j.spa.2016.09.013.
    [17] S. Gautier and L. Thibault, Viability for constrained stochastic differential equations, Differ. Integral Equ., 6 (1993), 1395-1414. 
    [18] D. Goreac, Viability, invariance and reachability for controlled piecewise deterministic Markov processes associated to gene networks, ESAIM Control Optim. Calc. Var., 18 (2012), 401-426.  doi: 10.1051/cocv/2010103.
    [19] M. Huang, P. E. Caines and R. P. Malhame, Individual and mass behaviour in large population stochastic wireless power control problems: centralized and nash equilibrium solutions, in $42^{nd}$ IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475), 1 (2003), 98-103. doi: 10.1109/TAC.2004.835388.
    [20] M. Kac, Foundations of kinetic theory, Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, Volume 3: Contributions to Astronomy and Physics, (1956), 171-197.
    [21] J. Lasry and P. L. Lions., Mean field games, JPN. J. MATH., 2 (2007), 229-260.  doi: 10.1007/s11537-007-0657-8.
    [22] S. Méléard, Asymptotic behaviour of some interacting particle systems; McKean-Vlasov and Boltzmann models, Springer Berlin Heidelberg, Berlin, Heidelberg, (1996), 42-95
    [23] M. Nagumo, Uber die lage derintegralkurven gewhnlicher differentialgleichungen, Proc. Phys. Math. Soc. Japan, 24 (1942), 551-559. 
    [24] T. Nie and A. Rascanu, Deterministic characterization of viability for stochastic differential equation driven by fractional brownian motion, ESAIM Control Optim. Calc. Var., 18 (2012), 915-929.  doi: 10.1051/cocv/2011188.
    [25] P. Dai Pra and F. den Hollander, McKean-Vlasov limit for interacting random processes in random media, J. Stat. Phys., 84 (1996), 735-772.  doi: 10.1007/BF02179656.
    [26] M. Shaked and J. G. Shantikumar, Stochastic Orders, Springer Series in Statistics, Springer, New York, NY, 2007. doi: 10.1007/978-0-387-34675-5.
    [27] A. S. Sznitman, Nonlinear reflecting diffusion process, and the propagation of chaos and fluctuations associated, J. Funct. Anal., 56 (1984), 311-336.  doi: 10.1016/0022-1236(84)90080-6.
    [28] A. S. Sznitman, Topics in propagation of chaos, Ecole d'Eté de Probabilités de Saint-Flour XIX-1989, Springer Berlin Heidelberg, Berlin, Heidelberg, (1991), 165-251. doi: 10.1007/BFb0085169.
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