A Note on $G$- Optimal Stopping Problems

TL;DR

This paper investigates $G$-optimal stopping problems, establishing their well-definedness and providing a verification theorem, showing their equivalence to classical problems under certain conditions.

Contribution

It introduces a framework for $G$-optimal stopping problems, proving well-definedness and a verification theorem, and clarifies their relation to classical problems.

Findings

Established the quasi-continuity of the stopped process under $G$-expectation.

Proved a verification theorem for $G$-optimal stopping problems.

Showed equivalence to classical problems with convex or concave payoffs.

Abstract

We consider a class of discretionary stopping problems within the $G$-framework. We first establish the well-definedness of the stopping problem under the $G$-expectation, by showing the quasi-continuity of the stopped process. We then prove a verification theorem for $G$-optimal stopping problem. One corollary is a direct proof for the well-known fact that the $G$-optimal stopping problem is the same as the classical optimal stopping problem with appropriate parameters, when the payoff function is concave or convex.