Equilibrium concepts for time‐inconsistent stopping problems in continuous time

Abstract

Abstract 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.