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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. 

This article introduces a novel numerical approach, based on finite-volume techniques, for studying fully nonlinear coagulation–fragmentation models, where both the coagulation and fragmentation components of the collision operator are nonlinear. The models come from three-wave kinetic equations, a pivotal framework in wave turbulence theory. Despite the importance of wave turbulence theory in physics and mechanics, there have been very few numerical schemes for three-wave kinetic equations, in which no additional assumptions are manually imposed on the evolution of the solutions, and the current manuscript provides one of the first of such schemes. To the best of our knowledge, this also is the first numerical scheme capable of accurately capturing the long-term asymptotic behaviour of solutions to a fully nonlinear coagulation–fragmentation model. The scheme is implemented on some test problems, demonstrating strong alignment with theoretical predictions of energy cascade rates, rigorously obtained in the work (Soffer & Tran. 2020Commun. Math. Phys.376, 2229–2276. (doi:10.1007/BF01419532)). We further introduce a weighted finite-volume variant to ensure energy conservation across varying degrees of kernel homogeneity. Convergence and first-order consistency are established through theoretical analysis and verified by experimental convergence orders in test cases. 

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. 

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].