Dynamic optimal control for distress large financial networks and Mean field systems with jumps ; Contrôle stochastique de systèmes à champs moyens et applications au contrôle du risque systémique

This thesis presents models and methodologies to understand the control of systemic risk in large systems. We propose two approaches. The first one is structural: a financial system is represented as a network of institutions. They have strategic interactions as well as direct interactions through linkages in a contagion process. The novelty of our approach is that these two types of interactions are intertwined themselves and we propose new notions of equilibria for such games and analyze the systemic risk emerging in equilibrium. The second approach is a reduced form. We model the dynamics of regulatory capital using a mean field operator: required capital depends on the standalone risk but also on the evolution of the capital of all other banks in the system. In this model, required capital is a dynamic risk measure and is represented as the solution of a mean-field BDSE with jumps. We show a novel dual representation theorem. In the context of mean-field BSDEs the representation gives yield to a stochastic discount factor and a worst-case probability measure that encompasses the overall interactions in the system. We also solve the optimal stopping problem of dynamic risk measure by connecting it to the solution of reflected mean-field BSDE with jumps. Finally, We provide a comprehensive model for the order book dynamics and optimal Market making strategy appeared in liquidity risk problems ; Cette thèse propose des modèles et des méthodes pour étudier le contrôle du risque dans de larges systèmes financiers. Nous proposons dans une première partie une approche structurelle : nous considérons un système financier représenté comme un réseau d’institutions connectées entre elles par des interactions stratégiques sources de financement mais également par des interactions qui les exposent à un risque de contagion de défaut. La nouveauté de notre approche réside dans le fait que ces deux types d’interaction interfèrent. Nous proposons des nouvelles notions d’équilibre pour ces systèmes et étudions la connectivité ...