Conditional Optimal Stopping: A Time-Inconsistent Optimization

Annals of Applied Probability, 30(4), 1669-1692, 2020

34 Pages Posted: 25 Jun 2019 Last revised: 6 Nov 2021

See all articles by Marcel Nutz

Marcel Nutz

Columbia University

Yuchong Zhang

University of Toronto - Department of Statistics

Date Written: January 17, 2019

Abstract

Inspired by recent work of P.-L. Lions on conditional optimal control, we introduce a problem of optimal stopping under bounded rationality: the objective is the expected payoff at the time of stopping, conditioned on another event. For instance, an agent may care only about states where she is still alive at the time of stopping, or a company may condition on not being bankrupt. We observe that conditional optimization is time-inconsistent due to the dynamic change of the conditioning probability and develop an equilibrium approach in the spirit of R. H. Strotz’ work for sophisticated agents in discrete time. Equilibria are found to be essentially unique in the case of a finite time horizon whereas an infinite horizon gives rise to non-uniqueness and other interesting phenomena. We also introduce a theory which generalizes the classical Snell envelope approach for optimal stopping by considering a pair of processes with Snell-type properties.

Keywords: Conditional optimal stopping, time-inconsistency, equilibrium

Suggested Citation

Nutz, Marcel and Zhang, Yuchong, Conditional Optimal Stopping: A Time-Inconsistent Optimization (January 17, 2019). Annals of Applied Probability, 30(4), 1669-1692, 2020, Available at SSRN: https://ssrn.com/abstract=3409585 or http://dx.doi.org/10.2139/ssrn.3409585

Marcel Nutz

Columbia University ( email )

Yuchong Zhang (Contact Author)

University of Toronto - Department of Statistics ( email )

700 University Ave.
Toronto, Ontario M5S 1Z5
Canada

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