2023/05/06 by Alain Bensoussan, Ho Man Tai, Bensoussan, Alain +3
Economics, Econometrics and Finance · Mathematics · #49J55 #49K45 #49N80 #60H30 #93E20 #Analysis of PDEs (math.AP) #FOS: Mathematics #Gas Dynamics and Kinetic Theory #Mathematical Biology Tumor Growth #Optimization and Control (math.OC) #Probability (math.PR) #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.2305.04019
openalex publication_date 2023/05/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this article, we provide an original systematic global-in-time analysis of mean field type control problems on ℝn with generic cost functionals by the modified approach but not the same, firstly proposed in [7], as the ``lifting'' idea introduced by P. L. Lions. As an alternative to the recent popular analytical method by tackling the master equation, we resolve the control problem in a certain proper Hilbert subspace of the whole space of L2 random variables, it can be regarded as tangent space attached at the initial probability measure. The present work also fills the gap of the global-in-time solvability and extends the previous works of [7,11] which only dealt with quadratic cost functionals in control; 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 regularities of the value function, including its linearly functional differentiability, which leads to the classical well-posedness 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 well-posedness of the master equation.