Authors’ abstract: The game approach to the theory of optimal stopping assumes two players, the “controller” and the “stopper”. The reward of the game is a nonnegative process \(Y\) with RCLL paths on a time-horizon \([0,T]\) of finite length, and is adapted to a filtration which satisfies the usual conditions. The controller is given a choice from a set of possible models in the form of a family of probability measures, all of which are equivalent to a reference probability \(Q\) on a given measurable space. The stopper maximizes his expected reward by choosing an optimal \(F\)-stopping time.
We explore two types of problems, a cooperative and a non-cooperative game, as the controller may work in collaboration or in competition with the stopper. We calculate the maximum expected reward \(R=\sup_{P\in ?}\sup_{T\in ?}E^P(Y_T)\) of the cooperative game, and find necessary and/or sufficient conditions for the existence of an optimal stopping time \(T^*\) and of an optimal model \(P^*\). Then we study the stochastic game with upper value and lower value, we state conditions under which this game has value and conditions under which there exists a saddle-point \((T^*,P^*)\) of strategies. We also present an application of these games to the pricing American options under constraints.