Optimal stopping: Bermudan strategies meet non-linear evaluations

TL;DR

This paper develops a theoretical framework for optimal stopping problems involving Bermudan strategies and non-linear evaluations, providing characterization, dynamic programming, and martingale properties under general conditions.

Contribution

It introduces a novel ($ ho$,$ heta$)-Snell envelope and establishes a dynamic programming principle for non-linear evaluations in Bermudan stopping problems.

Findings

Characterization of the value family V via ($ ho$,$ heta$)-Snell envelope

Establishment of a dynamic programming principle for the problem

Identification of an optimality criterion through ($ ho$,$ heta$)-martingale property

Abstract

We address an optimal stopping problem over the set of Bermudan-type strategies $\Theta$ (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 $\rho$. We provide a characterization of the value family V in terms of what we call the ($\rho$,$\Theta$) -Snell envelope of the pay-off family. We establish a Dynamic Programming Principle. We provide an optimality criterion in terms of a ($\rho$,$\Theta$) -martingale property of V on a stochastic interval. We investigate the ($\rho$,$\Theta$)-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 $\omega$. We provide examples of non-linear operators entering our framework.