On Time-Inconsistent Extended Mean-Field Control Problems with Common Noise

References

• [1] Acciaio B, Backhoff-Veraguas J, Carmona R (2017) Extended mean field control problems: Stochastic maximum principle and transport perspective. SIAM J. Control Optim. 57(6):3666–3693.Crossref, Google Scholar
• [2] Andersson D, Djehiche B (2011) A maximum principle for SDEs of mean-field type. Appl. Math. Optim. 63:341–356.Crossref, Google Scholar
• [3] Bayraktar E, Wang Z (2025) On time‐inconsistency in mean‐field games. Math. Finance 35(3):613–635.Crossref, Google Scholar
• [4] Bayraktar E, Cosso A, Pham H (2018) Randomized dynamic programming principle and Feynman-Kac representation for optimal control of McKean-Vlasov dynamics. Trans. Amer. Math. Soc. 370:2115–2160.Crossref, Google Scholar
• [5] Bensoussan A, Frehse J, Yam SCP (2015) The master equation in mean field theory. J. Mathématiques Pures Appliquées 103(6):1441–1474.Crossref, Google Scholar
• [6] Björk T, Khapko M, Murgoci A (2017) On time-inconsistent stochastic control in continuous time. Finance Stochastics 21:331–360.Crossref, Google Scholar
• [7] Bo L, Wang J, Wei X, Yu X (2024) Extended mean-field control problems with Poissonian common noise: Stochastic maximum principle and Hamiltonian-Jacobi-Bellman equation. Preprint, submitted July 7, https://arxiv.org/abs/2407.05356.Google Scholar
• [8] Buckdahn R, Djehiche B, Li J (2011) A general stochastic maximum principle for SDEs of mean-field type. Appl. Math. Optim. 64:197–216.Crossref, Google Scholar
• [9] Buckdahn R, Li J, Ma J (2017) A mean-field stochastic control problem with partial observations. Ann. Appl. Probab. 27(5):3201–3245.Crossref, Google Scholar
• [10] Buckdahn R, Li J, Li J, Xing C (2025) Path-dependent controlled mean-field coupled forward-backward SDEs: The associated stochastic maximum principle. SIAM J. Control Optim. 63(3):2124–2153.Crossref, Google Scholar
• [11] Buckdahn R, Li J, Peng S, Rainer C (2017) Mean-field stochastic differential equations and associated PDEs. Ann. Probab. 45(2):824–878.Crossref, Google Scholar
• [12] Cai H, Chen D, Peng Y, Wei W (2022) On the time-inconsistent deterministic linear-quadratic control. SIAM J. Control Optim. 60(2):968–991.Crossref, Google Scholar
• [13] Cardaliaguet P (2013) Notes on mean field games. Notes from P.-L. Lions' Lectures, Collège de France, Paris.Google Scholar
• [14] Carmona R, Delarue F (2015) Forward–backward stochastic differential equations and controlled McKean–Vlasov dynamics. Ann. Probab. 43(5):2647–2700.Crossref, Google Scholar
• [15] Carmona R, Zhu X (2016) A probabilistic approach to mean field games with major and minor players. Ann. Appl. Probab. 26(3):1535–1580.Crossref, Google Scholar
• [16] Chassagneux JF, Crisan D, Delarue F (2022) A Probabilistic Approach to Classical Solutions of the Master Equation for Large Population Equilibria, Memoirs of the American Mathematical Society, vol. 280 (American Mathematical Society, Providence, RI), 1379.Google Scholar
• [17] Ekeland I, Pirvu TA (2008) Investment and consumption without commitment. Math. Financial Econom. 2(1):57–86.Crossref, Google Scholar
• [18] Ekeland I, Mbodji O, Pirvu TA (2012) Time-consistent portfolio management. SIAM J. Financial Math. 3(1):1–32.Crossref, Google Scholar
• [19] Friedman A (1964) Partial Differential Equations of Parabolic Type (Robert Krieger Publishing Company, Malabar, FL).Google Scholar
• [20] He X, Jiang Z (2021) On the equilibrium strategies for time-inconsistent problems in continuous time. SIAM J. Control Optim. 59(5):3860–3886.Crossref, Google Scholar
• [21] Hu Y, Jin H, Zhou XY (2012) Time-inconsistent stochastic linear–quadratic control. SIAM J. Control Optim. 50(3):1548–1572.Crossref, Google Scholar
• [22] Hu Y, Jin H, Zhou XY (2017) Time-inconsistent stochastic linear-quadratic control: Characterization and uniqueness of equilibrium. SIAM J. Control Optim. 55(2):1261–1279.Crossref, Google Scholar
• [23] Ladyženskaja OA, Solonnikov VA, Ural’ceva NN, Ladyženskaja OA (1968) Linear and Quasi-Linear Equations of Parabolic Type, Translations of Mathematical Monographs, vol. 23 (American Mathematical Society, Providence, RI).Crossref, Google Scholar
• [24] Laurière M, Pironneau O (2016) Dynamic programming for mean-field type control. J. Optim. Theory Appl. 169:902–924.Crossref, Google Scholar
• [25] Lei Q, Pun CS (2023) Nonlocal fully nonlinear parabolic differential equations arising in time-inconsistent problems. J. Differential Equations 358:339–385.Crossref, Google Scholar
• [26] Lei Q, Pun CS (2024) Nonlocality, nonlinearity, and time inconsistency in stochastic differential games. Math. Finance 34(1):190–256.Crossref, Google Scholar
• [27] Li J (2012) Stochastic maximum principle in the mean-field controls. Automatica 48(2):366–372.Crossref, Google Scholar
• [28] Lü Q, Ma B (2024) Time-inconsistent linear quadratic optimal control problem for forward-backward stochastic differential equations. ESAIM: COCV 30:76.Crossref, Google Scholar
• [29] Mei H, Zhu C (2020) Closed-loop equilibrium for time-inconsistent McKean–Vlasov controlled problem. SIAM J. Control Optim. 58(6):3842–3867.Crossref, Google Scholar
• [30] Ni YH, Zhang JF, Krstic M (2017) Time-inconsistent mean-field stochastic LQ problem: Open-loop time-consistent control. IEEE Trans. Automatic Control 63(9):2771–2786.Google Scholar
• [31] Ni YH, Li X, Zhang JF, Krstic M (2019) Mixed equilibrium solution of time-inconsistent stochastic linear-quadratic problem. SIAM J. Control Optim. 57(1):533–569.Crossref, Google Scholar
• [32] Pham H, Wei X (2017) Dynamic programming for optimal control of stochastic McKean–Vlasov dynamics. SIAM J. Control Optim. 55(2):1069–1101.Crossref, Google Scholar
• [33] Pham H, Wei X (2018) Bellman equation and viscosity solutions for mean-field stochastic control problem. ESAIM: COCV 24(1):437–461.Crossref, Google Scholar
• [34] Strotz RH (1955) Myopia and inconsistency in dynamic utility maximization. Rev. Econom. Stud. 23(3):165–180.Crossref, Google Scholar
• [35] Wang T (2019) Characterizations of equilibrium controls in time inconsistent mean-field stochastic linear quadratic problems. I. Math. Control Related Fields 9(2):385–409.Crossref, Google Scholar
• [36] Wang T, Yong J (2019) Backward stochastic Volterra integral equations—Representation of adapted solutions. Stochastic Processes Appl. 129(12):4926–4964.Crossref, Google Scholar
• [37] Wei Q, Yong J, Yu Z (2017) Time-inconsistent recursive stochastic optimal control problems. SIAM J. Control Optim. 55(6):4156–4201.Crossref, Google Scholar
• [38] Wonham WM (1968) On a matrix Riccati equation of stochastic control. SIAM J. Control Optim. 6(4):681–697.Crossref, Google Scholar
• [39] Wu C, Zhang J (2020) Viscosity solutions to parabolic master equations and McKean-Vlasov SDEs with closed-loop controls. Ann. Appl. Probab. 30(2):936–986.Crossref, Google Scholar
• [40] Yan W, Yong J (2019) Time-inconsistent optimal control problems and related issues. Yin G, Zhang Q, eds. Modeling, Stochastic Control, Optimization, and Applications (Springer International Publishing, Cham), 533–569.Crossref, Google Scholar
• [41] Yong J (2012) Time-inconsistent optimal control problems and the equilibrium HJB equation. Math. Control Related Fields 2(3):271–329.Crossref, Google Scholar
• [42] Yong J (2013) Linear-quadratic optimal control problems for mean-field stochastic differential equations. SIAM J. Control Optim. 51(4):2809–2838.Crossref, Google Scholar
• [43] Yong J (2017) Linear-quadratic optimal control problems for mean-field stochastic differential equations—Time-consistent solutions. Trans. Amer. Math. Soc. 377(7):5467–5523.Google Scholar
• [44] Yong J, Zhou XY (1999) Stochastic Controls: Hamiltonian Systems and HJB Equations (Stochastic Modelling and Applied Probability), vol. 43 (Springer, New York).Crossref, Google Scholar
• [45] Yu X, Yuan F (2026) Time-inconsistent mean-field stopping problems: A regularized equilibrium approach. Finance Stochastics 30:179–236.Crossref, Google Scholar
• [46] Zhou J, Touzi N, Zhang J (2024) Viscosity solutions for HJB equation on the process space: Applications to mean field control with common noise. Preprint, submitted January 10, https://arxiv.org/abs/2401.04920.Google Scholar