Optimal stopping: Bermudan strategies meet non-linear evaluations

Arrêt optimal: une rencontre entre les stratégies Bermudiennes et les évaluations non-linéaires

Miryana Grigorova (1) , Marie-Claire Quenez (2) , Peng Yuan (1)

1 University of Warwick [Coventry]
2 UFR Mathématiques et informatique [Sciences] - Université Paris Cité

We address an optimal stopping problem over the set of Bermudan-type strategies Θ (which we understand in a more general sense than the stopping strategies for Bermudan options in finance) and with non-linear operators (non-linear evaluations) assessing the rewards, under general assumptions on the non-linear operators ρ. We provide a characterization of the value family V in terms of what we call the (ρ,Θ) -Snell envelope of the pay-off family. We establish a Dynamic Programming Principle. We provide an optimality criterion in terms of a (ρ,Θ) -martingale property of V on a stochastic interval. We investigate the (ρ,Θ)-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 entering our framework.

Mots clés: optimal stopping; Bermudan stopping strategy; non-linear operator; non-linear evaluation; g-expectation; dynamic risk measure; dynamic programming principle; non-linear Snell envelope family; Bermudan strategy

Domaines: Optimisation et contrôle [math.OC]; Optimisation et contrôle [math.OC]; Probabilités [math.PR]

Miryana Grigorova, Marie-Claire Quenez, Peng Yuan. Optimal stopping: Bermudan strategies meet non-linear evaluations. 2023. ⟨hal-03938727⟩