Time-consistent stopping under decreasing impatience

• Published in: Finance and stochastics Vol. 22; no. 1; pp. 69 - 95
• Main Authors: Huang, Yu-Jui, Nguyen-Huu, Adrien
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.01.2018; Springer Nature B.V; Springer Verlag (Germany)
• Subjects: Discounting; Economic Theory/Quantitative Economics/Mathematical Methods; Economics; Economics and Finance; Equilibrium; Finance; Fixed points (mathematics); Functions (mathematics); Game theory; Humanities and Social Sciences; Inconsistency; Insurance; Iterative methods; Management; Mathematical analysis; Mathematics; Mathematics and Statistics; Operators (mathematics); Policies; Probability Theory and Stochastic Processes; Quantitative Finance; Statistics for Business; Optimal stopping; Hyperbolic discounting; 60G40; Time inconsistency; Subgame-perfect Nash equilibrium; G02; Decreasing impatience; 91B06; D90; Iterative approach; D81; C61; subgame-perfect Nash equilibrium; optimal stopping; time inconsistency; decreasing impatience; hyperbolic discounting; iterative approach
• ISSN: 0949-2984; 1432-1122
• DOI: 10.1007/s00780-017-0350-6

Under non-exponential discounting, we develop a dynamic theory for stopping problems in continuous time. Our framework covers discount functions that induce decreasing impatience. Due to the inherent time inconsistency, we look for equilibrium stopping policies, formulated as fixed points of an operator. Under appropriate conditions, fixed-point iterations converge to equilibrium stopping policies. This iterative approach corresponds to the hierarchy of strategic reasoning in game theory and provides “agent-specific” results: it assigns one specific equilibrium stopping policy to each agent according to her initial behavior. In particular, it leads to a precise mathematical connection between the naive behavior and the sophisticated one. Our theory is illustrated in a real options model.