# Conditional Optimal Stopping: A Time-Inconsistent Optimization

## TL;DR

This paper introduces a new framework for optimal stopping problems where the decision is conditioned on events like survival or avoiding bankruptcy, addressing time-inconsistency through equilibrium solutions.

## Contribution

It develops a novel equilibrium approach for conditional optimal stopping, extending classical methods and analyzing uniqueness and non-uniqueness in finite and infinite horizons.

## Key findings

- Equilibrium solutions are unique in finite horizon cases.
- Infinite horizon problems exhibit non-uniqueness and complex phenomena.
- Generalization of the Snell envelope for conditioned processes.

## 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.