Optimal stopping under model uncertainty in a general setting

We consider the optimal stopping time problem under model uncertainty, R(v) = ess\ sup_ P ∈ P ess\ sup_τ ∈ Sᵥ E^ P[Y(τ) Fᵥ], for every stopping time v, within the framework of families of random variables indexed by stopping times. This setting is more general than the classical setup of stochastic processes, notably allowing for general payoff processes that are not necessarily right-continuous. Under weaker integrability, with regularity assumptions for the reward family Y = (Y(v), v∈ S), the existence of an optimal stopping time is demonstrated. Sufficient conditions for the existence of an optimal model are then determined. For this purpose, we present a universal optional decomposition for the generalized Snell envelope family associated with Y. This decomposition is then employed to prove the existence of an optimal probability model and to study its properties.