Equilibrium concepts for time‐inconsistent stopping problems in continuous time

• Published in: Mathematical finance Vol. 31; no. 1; pp. 508 - 530
• Main Authors: Bayraktar, Erhan, Zhang, Jingjie, Zhou, Zhou
• Format: Journal Article
• Language: English
• Published: Oxford Blackwell Publishing Ltd 01.01.2021
• Subjects: Equilibrium; Finance; Game theory; Markov chains; mild equilibria; nonexponential discounting; optimal stopping; Optimization; strong equilibria; subgame perfect Nash equilibrium; time inconsistency; weak equilibria
• ISSN: 0960-1627; 1467-9965; 1467-9965
• DOI: 10.1111/mafi.12293

A new notion of equilibrium, which we call strong equilibrium, is introduced for time‐inconsistent stopping problems in continuous time. Compared to the existing notions introduced in Huang, Y.‐J., & Nguyen‐Huu, A. (2018, Jan 01). Time‐consistent stopping under decreasing impatience. Finance and Stochastics, 22(1), 69–95 and Christensen, S., & Lindensjö, K. (2018). On finding equilibrium stopping times for time‐inconsistent markovian problems. SIAM Journal on Control and Optimization, 56(6), 4228–4255, which in this paper are called mild equilibrium and weak equilibrium, respectively, a strong equilibrium captures the idea of subgame perfect Nash equilibrium more accurately. When the state process is a continuous‐time Markov chain and the discount function is log subadditive, we show that an optimal mild equilibrium is always a strong equilibrium. Moreover, we provide a new iteration method that can directly construct an optimal mild equilibrium and thus also prove its existence.