Viscosity Solution Theory for Mean Field Stochastic Controls and Feynman-Kac Representation for Nonlinear Time-dependent Schrödinger Equations

Abstract

Mean field (McKean–Vlasov) control problems model strategic decision-making in large pop-ulations of (approximately) symmetric agents interacting through an aggregate, or the so called mean field term. The dependence on the population measure in both the state dy-namics and the objective naturally leads to infinite-dimensional formulations. As in finite- dimensional cases, the associated Hamilton–Jacobi–Bellman equations typically do not admit classical solutions, making viscosity solutions a natural and powerful framework. In the first part of this thesis, we explore the theory of viscosity solutions for mean field control problems. In the second part, we study a novel Feynman–Kac representation. Traditionally, Feyn-man–Kac formulas connect parabolic partial differential equations (PDEs) with stochastic differential equations (SDEs). We extend this connection to Schrödinger-type equations, which are prototypical non-parabolic PDEs, and demonstrate how this approach facilitates efficient simulation of high-dimensional PDEs.