Title: Conditional Optimal Stopping: A Time-Inconsistent Optimization,
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.