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

49 Pages Posted: 10 Jan 2019 Last revised: 18 Sep 2020

See all articles by Xue Dong He

Xue Dong He

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

Zhao Li Jiang

The Chinese University of Hong Kong (CUHK) - Department of Systems Engineering and Engineering Management; National University of Singapore (NUS) - Risk Management Institute

Date Written: January 30, 2019

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, Zhao Li, On the Equilibrium Strategies for Time-Inconsistent Problems in Continuous Time (January 30, 2019). 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

Zhao Li Jiang

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

Hong Kong
China

National University of Singapore (NUS) - Risk Management Institute ( email )

21 Heng Mui Keng Terrace
Level 4
Singapore, 119613
Singapore

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