We introduce a framework for optimal stopping which could be seen as lying between the discrete and the continuous time. We call the set of stopping strategies available to the agent the set of Bermudan-type strategies Θ (which are to be understood in a more general sense than the stopping strategies for Bermudan options in finance). We address the optimal stopping problem under general assumptions on the non-linear operators ρ assessing the rewards (ρ might depend on two time indices: the time of evaluation and the horizon). We provide a characterization of the value family V in terms of what we call the (Θ, ρ)-Snell envelope of the pay-off family. We establish a Dynamic Programming Principle. We provide an optimality criterion in terms of a (Θ, ρ)-martingale property of V on a stochastic interval. We investigate the (Θ, ρ)-martingale structure and we show that the “first time” when the value family coincides with the pay-off family is optimal. The reasoning simplifies in the case where there is a finite number n of pre-described stopping times, where n does not depend on the scenario ω. We provide examples of non-linear operators which enter our framework.

If time permits, we will also consider a multiple optimal stopping problem (where the agent has multiple exercise times) and we will tackle the problem by generalising the so-called reduction approach (well-known in the case of linear expectations). We will also mention some results on non-zero-sum non-linear games of stopping in the above framework.

The talk is based on joint works with Marie-Claire Quenez (Paris) and Peng Yuan (Warwick).