Osborne, Yohance Andre Peter; (2024) Analysis and numerical approximation of mean field game partial differential inclusions. Doctoral thesis (Ph.D), UCL (University College London).

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Abstract

Mean Field Games (MFG) are models for Nash equilibria of large-population differential games of stochastic optimal control. In many applications, the MFG model is described by a non-linear system of partial differential equations (PDE) which consists of a Hamilton—Jacobi—Bellman (HJB) equation for the value function of each player and a Kolmogorov—Fokker—Planck (KFP) equation for the player distribution. The standard formulation of the MFG PDE system requires that the Hamiltonian be a differentiable function of the generalised momentum to determine the advective term in the KFP equation. However, the structure of the underlying optimal control problem can lead to Hamiltonians that are convex but non-differentiable. In this thesis, we develop the analysis and numerical analysis of a general class of second-order MFG systems with convex, Lipschitz but (possibly) non-differentiable Hamiltonians in both stationary and time-dependent settings. In particular, we introduce a modelling framework that proposes the generalisation of the MFG system to a Partial Differential Inclusion (PDI) based on interpreting the derivative of the Hamiltonian in terms of the Moreau—Rockafellar subdifferential for convex functions. We prove the existence of weak solutions to MFG PDI under mild conditions on the model data, prior to establishing the uniqueness of weak solutions under a monotonicity condition like the one considered by Lasry and Lions. Moreover, we propose a class of monotone stabilised finite element methods (FEMs) for MFG PDI that is based on conforming piecewise affine finite element spaces for the spatial discretisation. We prove that the methods from this class possess strong convergence properties towards weak solutions to MFG PDI in suitable norms; we also show that these properties are stronger in the special case of differentiable Hamiltonians. We conduct numerical experiments that highlight the ability of the method to approximate non-smooth weak solutions at optimal convergence rates.

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