Switching Identities by Probabilistic Means

TL;DR

This paper introduces a simple probabilistic approach to switching identities in potential theory and stochastic analysis, revealing new symmetries between solutions to the Skorokhod embedding problem and connecting optimal stopping problems.

Contribution

It provides a novel probabilistic proof of switching identities, previously derived through analytic methods, and uncovers a new symmetry between Root and Rost solutions to the SEP.

Findings

Probabilistic proof of switching identities

Uncovered symmetry between Root and Rost solutions

Connected optimal stopping and Skorokhod embedding problems

Abstract

Switching identities have a long history in potential theory and stochastic analysis. In recent work of Cox and Wang, a switching identity was used to connect an optimal stopping problem and the Skorokhod embedding problem (SEP). Typically switching identies of this form are derived using deep analytic connections. In this paper, we prove the switching identities using a simple probabilistic argument, which furthermore highlights a previously unexplored symmetry between the Root and Rost solutions to the SEP.