Optimal Stopping under Model Uncertainty

Optimal Stopping under Model Uncertainty Ingrid-Mona Zamfirescu The aim of this paper is to extend the theory of optimal stopping to cases in which there is model-uncertainty. This means that we are given a set of possible models in the form of a family P of probability measures, equivalent to a reference probability measure Q on a given measurable space (Ω,F). We are also given a filtration F = {Ft}t≥0 that satisfies the “usual conditions”, and a nonnegative adapted reward process Y with RCLL paths. We shall denote by S the class of F− stopping times. Our goal is to compute the maximum expected reward under the specified model uncertainty, i.e., to calculate R = supP∈P supτ∈S E P (Yτ ), and to find necessary and/or sufficient conditions for the existence of an optimal stopping time τ ∗ and an optimal model P ∗. We also study the stochastic game with the upper value V = infP∈P supτ∈S E P (Yτ ) and the lower value V = supτ∈S infP∈P E P (Yτ ); we state conditions under which this game has value, i.e. V = V =: V , and conditions under which there exists a “saddle-point” (τ ∗, P ∗) of strategies, i.e. V = E ∗ (Yτ∗).