A Semi-Markov Dynamic Capital Injection Problem for Distressed Banks

Our study investigates the optimal dividend strategy for a bank, taking into account the potential for government capital injections. We explore different types of government interventions, such as liberal, transparent, or uncertain strategies, and consider both single and multiple types of interventions. Our approach differs from others as it focuses on interventions that aim to maintain the overall stability of the financial system, rather than just addressing banks that have already sought government assistance or are in dire need of it. Specifically, we focus on situations where the government is more likely to assist banks that have not requested its intervention or that are not too difficult to save. To accomplish this, we conduct a comprehensive examination of all possible scenarios involving a single, one-time capital injection and derive explicit solutions for the associated optimal control problem. Furthermore, we expand the model to include semi-Markov dynamic capital injection processes and show that the optimal control is the unique viscosity solution of a Hamilton–Jacobi–Bellman equation. The government’s strategy also takes into account the bank’s solvency and any past government interventions

A Mean Field Capital Accumulation Game with HARA Utility

This paper introduces a mean field modeling framework for consumption-accumulation optimization. The production dynamics are generalized from stochastic growth theory by addressing the collective impact of a large population of similar agents on efficiency. This gives rise to a stochastic dynamic game with mean field coupling in the dynamics, where we adopt a hyperbolic absolute risk aversion (HARA) utility functional for the agents. A set of decentralized strategies is obtained by using the Nash certainty equivalence approach. To examine the long-term behavior we introduce a notion called the relaxed stationary mean field solution. The simple strategy computed from this solution is used to investigate the out-of-equilibrium behavior of the mean field system. Interesting nonlinear phenomena can emerge, including stable equilibria, limit cycles and chaos, which are related to the agent's sensitivity to the mean field

Essays on Business Analytics and Game Theory

Ph.D

Mean field capital accumulation games: The long time behavior

In (Huang, Dyn. Games AppJ., 2013) a mean field capital accumulation game with HARA utility was studied and by using a notion called the relaxed mean field solution it was shown that mean field dynamics in the closed-loop may exhibit stable equilibria or oscillatory (even chaotic) behavior. This paper analyzes the infinite horizon game while addressing the transient behavior of the mean field when the system dynamics can ensure predictable mean field behavior for rational agents. We analyze the associated optimal control and derive the fixed point equation for consistent mean field approximations. We further investigate numerical solutions to the coupled equation system characterizing the optimal response and the consistent mean field approximation