On the equilibrium strategies for time-inconsistent problems in continuous time

Abstract

In a continuous-time setting, the existing notion of equilibrium strategies for time-inconsistent problems in the literature, referred to as weak equilibrium, is not fully aligned with the standard definition of equilibrium in game theory in that the agent may be willing to deviate from a given weak equilibrium strategy. To address this issue, [Y.-J. Huang and Z. Zhou, Math. Oper. Res., 46 (2021), pp. 428--451] propose the notion of strong equilibrium for an infinite-time stochastic control problem in which an agent can control the generator of a time-homogeneous, continuous-time, finite-state Markov chain at each time. We study weak and strong equilibria in a general diffusion framework, provide necessary conditions for a strategy to be a strong equilibrium, and prove that strong equilibrium strategies do not exist for three investment and consumption problems. Finally, we propose a new notion of equilibrium strategies, referred to as regular equilibrium, show that it implies weak equilibrium, provide a sufficient condition under which a weak equilibrium strategy becomes a regular equilibrium, and show that this condition holds for many time-inconsistent problems.