More Like this

1. Mean field game master equations with anti-monotonicity conditions

   https://doi.org/10.4171/jems/1455  

   Mou, Chenchen; Zhang, Jianfeng (September 2025, Journal of the European Mathematical Society)

   It is well known that the monotonicity condition, either in Lasry–Lions sense or in displacement sense, is crucial for the global well-posedness of mean field game master equations, as well as for the uniqueness of mean field equilibria and solutions to mean field game systems. In the literature, the monotonicity conditions are always taken in a fixed direction. In this paper, we propose a new type of monotonicity condition in the opposite direction, which we call the anti-monotonicity condition, and establish the global well-posedness for mean field game master equations with non-separable Hamiltonians. Our anti-monotonicity condition allows our data to violate both the Lasry–Lions monotonicity and the displacement monotonicity conditions. 

   more » « less
2. Well-posedness of mean field games master equations involving non-separable local Hamiltonians

   https://doi.org/10.1090/tran/8760  

   Ambrose, David; Mészáros, Alpár (January 2023, Transactions of the American Mathematical Society)

   In this paper we construct short time classical solutions to a class of master equations in the presence of non-degenerate individual noise arising in the theory of mean field games. The considered Hamiltonians are non-separable andlocalfunctions of the measure variable, therefore the equation is restricted to absolutely continuous measures whose densities lie in suitable Sobolev spaces. Our results hold for smooth enough Hamiltonians, without any additional structural conditions as convexity or monotonicity. 

   more » « less
3. Mean field type control problems, some Hilbert-space-valued FBSDES, and related equations

   https://doi.org/10.1051/cocv/2025022  

   Bensoussan, Alain; Tai, Ho Man; Yam, Sheung_Chi Phillip (January 2025, ESAIM: Control, Optimisation and Calculus of Variations)

   In this article, we provide an original systematic global-in-time analysis of mean field type control problems on ℝⁿwith generic cost functions allowing quadratic growth by a novel “lifting” approach which is not the same as the traditional lifting. As an alternative to the recent popular analytical method of tackling master equations, we resolve the control problem in a proper Hilbert subspace of the whole space ofL²random variables, it can be regarded as a tangent space attached at the initial probability measure. The problem is linked to the global solvability of the Hilbert-space-valued forward–backward stochastic differential equation (FBSDE), which is solved by variational techniques here. We also rely on the Jacobian flow of the solution to this FBSDE to establish the regularity of the value function, including its linearly functional differentiability, which leads to the classical wellposedness of the Bellman equation. Together with the linear functional derivatives and the gradient of the linear functional derivatives of the solution to the FBSDE, we also obtain the classical wellposedness of the master equation. Our current approach imposes structural conditions directly on the cost functions. The contributions of adopting this framework in our study are twofold: (i) compared with imposing conditions on Hamiltonian, the structural conditions imposed in this work are easily verified, and less demanding on the cost functions while solving the master equation; and (ii) when the cost functions are not convex in the state variable or there is a lack of monotonicity of cost functions, an accurate lifespan can be provided for the local existence, which may not be that small in many cases. 

   more » « less
4. Mean field approach to stochastic control with partial information

   https://doi.org/10.1051/cocv/2021085  

   Bensoussan, Alain; Yam, Sheung Chi (January 2021, ESAIM: Control, Optimisation and Calculus of Variations)

   Buttazzo, G.; Casas, E.; de Teresa, L.; Glowinski, R.; Leugering, G.; Trélat, E.; Zhang, X. (Ed.)

   In our present article, we follow our way of developing mean field type control theory in our earlier works [Bensoussanet al., Mean Field Games and Mean Field Type Control Theory.Springer, New York (2013)], by first introducing the Bellman and then master equations, the system of Hamilton-Jacobi-Bellman (HJB) and Fokker-Planck (FP) equations, and then tackling them by looking for the semi-explicit solution for the linear quadratic case, especially with an arbitrary initial distribution; such a problem, being left open for long, has not been specifically dealt with in the earlier literature, such as Bensoussan [Stochastic Control of Partially Observable Systems. Cambridge University Press, (1992)] and Nisio [Stochastic control theory: Dynamic programming principle. Springer (2014)], which only tackled the linear quadratic setting with Gaussian initial distributions. Thanks to the effective mean-field theory, we propose a solution to this long standing problem of the general non-Gaussian case. Besides, our problem considered here can be reduced to the model in Bandiniet al.[Stochastic Process. Appl.129(2019) 674–711], which is fundamentally different from our present proposed framework. 

   more » « less
5. Maximum Principle for Mean Field Type Control Problems with General Volatility Functions

   Bensoussan, Alain; Huang, Ziyu; Yam, Sheung (July 2024, International game theory review)

   In this paper, we study the maximum principle of mean field type control problems when the volatility function depends on the state and its measure and also the control, by using our recently developed method in [Bensoussan, A., Huang, Z. and Yam, S. C. P. [2023] Control theory on Wasserstein space: A new approach to optimality conditions, Ann. Math. Sci. Appl.; Bensoussan, A., Tai, H. M. and Yam, S. C. P. [2023] Mean field type control problems, some Hilbert-space-valued FBSDEs, and related equations, preprint (2023), arXiv:2305.04019; Bensoussan, A. and Yam, S. C. P. [2019] Control problem on space of random variables and master equation, ESAIM Control Optim. Calc. Var. 25, 10]. Our method is to embed the mean field type control problem into a Hilbert space to bypass the evolution in the Wasserstein space. We here give a necessary condition and a sufficient condition for these control problems in Hilbert spaces, and we also derive a system of forward–backward stochastic differential equations. 

   more » « less