Game approach to the optimal stopping problem†

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 ∈ 𝒫 supτ ∈ 𝒮 E P (Y τ) of the cooperative game, and find necessary and/or sufficient conditions for the existence of an optimal stopping time τ* 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, i.e. and conditions under which there exists a saddle-point (τ*,P *) of strategies, i.e. We also present an application of these games to the pricing American options under constraints.