Expected Supremum Representation and Optimal Stopping

TL;DR

This paper develops an explicit integral representation for the value of optimal stopping problems of linear diffusions using expected supremum, connecting it with known principles like smooth fit and optimal stopping signals.

Contribution

It introduces a novel integral representation of the value function based on the joint distribution of extremal processes, linking it with the generator's monotonicity.

Findings

Explicit integral representation of the value function.

Connection with smooth fit principle and optimal stopping signals.

Illustrations in financial applications.

Abstract

We consider the representation of the value of an optimal stopping problem of a linear diffusion as an expected supremum of a known function. We establish an explicit integral representation of this function by utilizing the explicitly known joint probability distribution of the extremal processes. We also delineate circumstances under which the value of a stopping problem induces directly this representation and show how it is connected with the monotonicity of the generator. We compare our findings with existing literature and show, for example, how our representation is linked to the smooth fit principle and how it coincides with the optimal stopping signal representation. The intricacies of the developed integral representation are explicitly illustrated in various examples arising in financial applications of optimal stopping.