Integral equations for Rost's reversed barriers: existence and uniqueness results

TL;DR

This paper proves the existence and uniqueness of solutions to integral equations characterizing Rost's reversed barriers, extending previous results and applying to atom-less target distributions in the Skorokhod embedding problem.

Contribution

It establishes the unique solution of nonlinear integral equations for Rost's reversed barriers, generalizing known equations and surpassing existing uniqueness results.

Findings

Boundaries are the unique monotonic solutions to the integral equations.

Results apply to atom-less target distributions in the Skorokhod problem.

The integral equations generalize those in optimal stopping literature.

Abstract

We establish that the boundaries of the so-called Rost's reversed barrier are the unique couple of left-continuous monotonic functions solving a suitable system of nonlinear integral equations of Volterra type. Our result holds for atom-less target distributions $\mu$ of the related Skorokhod embedding problem. The integral equations we obtain here generalise the ones often arising in optimal stopping literature and our proof of the uniqueness of the solution goes beyond the existing results in the field.