| 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 ( W,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∈Psup t∈S EP ( Yt ), and to find necessary and/or sufficient conditions for the existence of an optimal stopping time t * and an optimal model P*. We also study the stochastic game with the upper value V¯ = infP∈Psup t∈S EP ( Yt ) and the lower value V&barbelow; = supt∈Sinf P∈P EP( Yt ); we state conditions under which this game has value, i.e. V¯ = V&barbelow; =: V, and conditions under which there exists a "saddle-point" ( t *, P*) of strategies, i.e. V = EP*( Yt* ). |
