On the Equilibrium Strategies for Time-Inconsistent Problems in Continuous Time

29 Pages Posted: 10 Jan 2019 Last revised: 7 Jul 2021

See all articles by Xue Dong He

Xue Dong He

The Chinese University of Hong Kong - Department of Systems Engineering and Engineering Management

Zhaoli Jiang

Hong Kong Polytechnic University

Date Written: July 7, 2021

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 the game theory in that the agent may be willing to deviate from a given weak equilibrium strategy. To address this issue, Huang and Zhou (2019, forthcoming in Mathematics of Operations Research) 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 equilibrium 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 four 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.

Keywords: stochastic control, time inconsistency, continuous-time setting, equilibrium strategies, portfolio selection

JEL Classification: C61, C73

Suggested Citation

He, Xue Dong and Jiang, Zhaoli, On the Equilibrium Strategies for Time-Inconsistent Problems in Continuous Time (July 7, 2021). Available at SSRN: https://ssrn.com/abstract=3308274 or http://dx.doi.org/10.2139/ssrn.3308274

Xue Dong He (Contact Author)

The Chinese University of Hong Kong - Department of Systems Engineering and Engineering Management ( email )

505 William M.W. Mong Engineering Building
The Chinese University of Hong Kong, Shatin, N.T.
Hong Kong
Hong Kong

HOME PAGE: http://https://sites.google.com/site/xuedonghepage/home

Zhaoli Jiang

Hong Kong Polytechnic University ( email )

Hung Hom
Kowloon
Hong Kong

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