Problems in Mathematical Finance Related to Time-inconsistency and Mean Field Games

Abstract

This thesis consists of two problems on time inconsistency and one problem on mean field games, all featuring the study of equilibrium and applications in economics and finance. In Chapter II, we deal with time inconsistency in the infinite horizon mean- variance stopping problem under discrete time setting. In order to determine a proper time-consistent plan, we investigate subgame perfect Nash equilibria among three different types of strategies, pure stopping times, randomized stopping times and liquidation strategies. We show that equilibria among pure stopping times or randomized stopping times may not exist, while an equilibrium liquidation strategy always exists. Furthermore, we argue that the mean-standard deviation variant of this problem makes more sense for this type of strategies in terms of time consistency. The existence and uniqueness of optimal equilibrium liquidation strategies are also analyzed. In Chapter III, we delve into equilibrium concepts for time inconsistent stopping problems in continuous time. We point out that the two existing notions of equi- librium in the literature, which we call mild equilibrium and weak equilibrium, are inadequate to capture the idea of subgame perfect Nash equilibrium. To characterize it more accurately, we introduce a new notion, strong equilibrium. It is proved that an optimal mild equilibrium is always a strong equilibrium. Moreover, we provide a new iteration method that can directly construct an optimal mild equilibrium and thus also guarantees its existence. xi In Chapter IV, we adopt a mean field game (MFG) approach to analyze a costly job search model with incomplete credit and insurance markets. The MFG approach enables us to quantify the impact of a class of countercyclical unemployment benefit policies on labor supply in general equilibrium. Our model provides two interesting predictions. First, the difference between unemployment rates under a countercyclical policy and an acyclical policy is positive and increases rapidly with the size of the aggregate shock. Second, compared with a baseline policy without means test, a means-tested policy which is targeted to provide more generous benefits to liquidity constrained individuals turns out to provide improved consumption insurance to all individuals as well as results in a lower equilibrium unemployment rate relative to a comparable non-targeted policy.