Viscosity Solutions to Hamilton-Jacobi Equations and Mean Field Game Theory for Systemic Risk Analysis in Nigeria

This dissertation develops original contributions to the theory of viscosity solutions for Hamilton-Jacobi-Bellman equations arising in mean field game formulations of systemic risk in large financial systems, with application to the modelling of coordinated bank run dynamics and contagion in the Nigerian banking sector. Mean field game theory, which models the strategic interactions of a continuum of rational agents as a coupled system of a backward Hamilton-Jacobi-Bellman equation and a forward Fokker-Planck equation, provides a mathematically tractable large-population limit for financial network models and offers novel analytical insights into systemic risk dynamics. The dissertation proves new uniqueness and stability theorems for viscosity solutions of second-order Hamilton-Jacobi-Bellman equations with quadratic Hamiltonian growth in the gradient variable, extending classical results to the degenerate parabolic case arising when agents have heterogeneous information structures. A master equation formulation of the mean field game is developed on the space of probability measures equipped with the Wasserstein metric, and well-posedness is established for a class of weakly coupled master equations relevant to financial contagion modelling. The mean field game equilibrium characterises the optimal liquidation strategies of banks facing a systemic liquidity shock, and the relationship between equilibrium behaviour and financial system fragility is characterised through the analysis of equilibrium uniqueness and multiplicity. Application to the 2016 Nigerian banking sector liquidity crisis calibrates the model using Central Bank of Nigeria supervisory data. Keywords: viscosity solutions, Hamilton-Jacobi equations, mean field games, systemic risk, Nigerian banking