Contagious McKean-Vlasov systems with positive feedback

We introduce a family of contagious McKean-Vlasov systems with positive feedback which are shown to arise as the mean field (or large population) limits of an associated family of finite particle systems. In the finite systems, each particle performs some variant of Brownian motion until it hits a lower barrier|at which point it is absorbed and the surviving particles are shifted down in the direction of the barrier. The latter may lead to further absorptions and hence a positive feedback loop is created that can model contagious feedback effects from hitting the barrier. More specifically, we propose to use this class of McKean-Vlasov systems as a simple framework for macroscopic modelling of systemic risk in large financial systems. In this case, each particle represents the distance-to-default of a financial institution, which is simply a measure of its financial ‘healthiness’ living on the positive half-line, and absorption at the origin then corresponds to default. The positive feedback emerges naturally by imposing that, whenever there is a default, all the other particles must be shifted closer to the origin|and thus closer to default through an endogenous notion of default contagion. In view of the above, the main contributions of this thesis are twofold. Firstly, we establish the convergence of the aforementioned finite dimensional particle systems to well-defined characterisations of their mean field limits. Secondly, we provide novel results on uniqueness and regularity for the appertaining class of limiting McKean-Vlasov problems.