Risk-Sensitive Mean Field Games with Common Noise: A Theoretical Study with Applications to Interbank Markets

In this paper, we address linear-quadratic-Gaussian (LQG) risk-sensitive mean field games (MFGs) with common noise. In this framework agents are exposed to a common noise and aim to minimize an exponential cost functional that reflects their risk sensitivity. We leverage the convex analysis method to derive the optimal strategies of agents in the limit as the number of agents goes to infinity. These strategies yield a Nash equilibrium for the limiting model. The model is then applied to interbank markets, focusing on optimizing lending and borrowing activities to assess systemic and individual bank risks when reserves drop below a critical threshold. We employ Fokker-Planck equations and the first hitting time method to formulate the overall probability of a bank or market default. We observe that the risk-averse behavior of agents reduces the probability of individual defaults and systemic risk, enhancing the resilience of the financial system. Adopting a similar approach based on stochastic Fokker-Planck equations, we further expand our analysis to investigate the conditional probabilities of individual default under specific trajectories of the common market shock.