On the Equilibrium Strategies for Time-Inconsistent Problems in Continuous Time
51 Pages Posted: 10 Jan 2019 Last revised: 31 Jan 2019
Date Written: January 30, 2019
In a continuous-time setting, the existing notion of equilibrium strategies for time-inconsistent problems in the literature, referred to as weak equilibria, is not fully aligned with the standard definition of equilibria in the game theory in that the agent may be willing to deviate from a given weak equilibrium strategy. To address this issue, we propose two new notions of equilibrium strategies for time-inconsistent problems, named regular equilibria and strong equilibria. For a large class of time-inconsistent problems, we derive sufficient conditions as well as necessary conditions for a strategy to be a regular equilibrium and to be a strong equilibrium. We examine three time-inconsistent portfolio selection problems in the literature and show that the weak equilibrium strategies derived therein are also regular equilibria but are not strong equilibria. We further provide an example to show that a weak equilibrium strategy may not be a regular equilibrium. Our results suggest that the notion of regular equilibria is preferred in the study of time-inconsistent problems.
Keywords: stochastic control, time inconsistency, continuous-time setting, equilibrium strategies, portfolio selection
JEL Classification: C61, C73
Suggested Citation: Suggested Citation