“…In some applications, however, information is a scarce resource, and the stopper then needs to base her decision only on the information available upon stopping. We study stopping problems of the type sup τ E g(τ, X τ , θ)1 {τ <γ } + h(γ, X γ , θ)1 {γ ≤τ } , (1) where X is a diffusion process; here g and h are given functions representing the payoff if stopping occurs before or after the random time horizon γ , respectively, and θ is a Bernoulli random variable representing the unknown state. This unknown state may influence the drift of the diffusion process X, the distribution of the random horizon γ and the payoff functions g and h.…”