# Expected Supremum Representation of a Class of Single Boundary Stopping Problems

## TL;DR

This paper presents an explicit integral representation for the value of certain optimal stopping problems involving linear diffusions, linking it to known probabilistic marginals and the smooth fit principle, with applications in finance.

## Contribution

It introduces a novel explicit integral representation of the value function for a class of optimal stopping problems, connecting it with the monotonicity of the generator and known probabilistic marginals.

## Key findings

- Explicit integral representation derived for the value function.
- Connection established between the representation and the smooth fit principle.
- Illustrations provided through financial application examples.

## Abstract

We consider the representation of the value of a class of optimal stopping problems of linear diffusions in a linearized form as an expected supremum of a known function. We establish an explicit integral representation of this representing function by utilizing the explicitly known marginals of the 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.