Optimal Multiple Stopping Problem under Nonlinear Expectation

TL;DR

This paper investigates the optimal multiple stopping problem within nonlinear expectations, introducing new methods for constructing optimal stopping times and representing them as hitting times under certain regularity conditions.

Contribution

It develops a novel approach to solve multiple stopping problems under nonlinear expectations, extending classical methods to more general reward families.

Findings

Constructed optimal stopping times for single and multiple problems.

Proved the value function can be interpreted via a new reward family.

Showed reward family and value functions can be aggregated as progressive processes.

Abstract

In this paper, we study the optimal multiple stopping problem under the filtration consistent nonlinear expectations. The reward is given by a set of random variables satisfying some appropriate assumptions rather than an RCLL process. We first construct the optimal stopping time for the single stopping problem, which is no longer given by the first hitting time of processes. We then prove by induction that the value function of the multiple stopping problem can be interpreted as the one for the single stopping problem associated with a new reward family, which allows us to construct the optimal multiple stopping times. If the reward family satisfies some strong regularity conditions, we show that the reward family and the value functions can be aggregated by some progressive processes. Hence, the optimal stopping times can be represented as hitting times.