Mean field type control problems, some Hilbert-space-valued FBSDES, and related equations

¹ International Center for Decision and Risk Analysis, Naveen Jindal School of Management, University of Texas at Dallas, Richardson, TX, USA
² School of Mathematical Sciences, Dublin City University, Dublin, Ireland
³ Department of Statistics, The Chinese University of Hong Kong, Hong Kong

^* Corresponding author: axb046100@utdallas.edu

Received: 19 April 2024
Accepted: 17 February 2025

Abstract

In this article, we provide an original systematic global-in-time analysis of mean field type control problems on ℝⁿ with 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 of L² random 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.