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The primary objective of this article is to present a general framework for users and applications of the master equation approach in extended mean field type control, for- mulated with a McKean-Vlasov stochastic differential equation that depends on the law of both the control and state variables. This control problem has recently gained significant attention and has been extensively studied at the level of the Bellman equa- tion. Here, we extend the analysis to the master equation and derive the corresponding Hamilton-Jacobi-Bellman equation. A key novelty of our approach is that we do not directly rely on the Fokker-Planck equation, which surprisingly leads to a significant simplification. We provide a concise theoretical presentation with proofs, as the stan- dard theory of stochastic control is not directly applicable. In the current work, the solution is constructed using an ansatz-based approach to dynamic programming via the master equation.We illustrate this method with a practical example. All proofs are presented in a self-contained manner. This paper offers a structured presentation of the extended mean field type control problem, serving as a valuable toolbox for users who are less focused on mathematical intricacies but seek a general framework for application. 

We develop a new dynamic continuous-time model of optimal consumption and investment to include independent stochastic labor income. We reduce the problem of solving the Bellman equation to a problem of solving an integral equation. We then explicitly characterize the optimal consumption and investment strategy as a function of income-to-wealth ratio. We provide some analytical comparative statics associated with the value function and optimal strategies. We also develop a quite general numerical algorithm for control iteration and solve the Bellman equation as a sequence of solutions to ordinary differential equations. This numerical algorithm can be readily applied to many other optimal consumption and investment problems especially with extra nondiversifiable Brownian risks, resulting in nonlinear Bellman equations. Finally, our numerical analysis illustrates how the presence of stochastic labor income affects the optimal consumption and investment strategy. Funding: A. Bensoussan was supported by the National Science Foundation under grant [DMS-2204795]. S. Park was supported by the Ministry of Education of the Republic of Korea and the National Research Foundation of Korea, South Korea [NRF-2022S1A3A2A02089950]. 

Many cities struggle with financing their infrastructure projects. When decision makers cannot fully capture the benefits of their investments, there is a risk of underinvestment. Hong Kong’s transit operator created a model where it not only collects fare revenues but also engages in property management, leveraging the positive effects of public transport on nearby property values. In the article titled “Monetizing Positive Externalities to Mitigate the Infrastructure Underinvestment Problem,” the authors present a stochastic Stackelberg game of timing to examine the reasoning behind this approach. The issue is complex because the operator faces a two-dimensional optimal stopping problem that cannot be simplified by changing the numéraire. The authors determine the operator’s optimal investment strategy through the use of a “penalized problem” and provide comparative statics. They also identify the conditions in which capitalizing on positive externalities can encourage infrastructure investment. Other management challenges share similar structures. 

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. 

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.