The Skorokhod problem in a time-dependent interval

TL;DR

This paper investigates the Skorokhod problem within a dynamic interval, providing existence, uniqueness, explicit solutions, and conditions affecting the local time variation, with applications to reflected Brownian motions.

Contribution

It offers new explicit formulas and conditions for the Skorokhod problem with moving boundaries, advancing understanding of reflected Brownian motion behavior.

Findings

Explicit solution formulas for the Skorokhod problem in time-dependent intervals

Conditions guaranteeing finite or infinite local time variation

Application to semimartingale property of reflected Brownian motions

Abstract

We consider the Skorokhod problem in a time-varying interval. We prove existence and uniqueness for the solution. We also express the solution in terms of an explicit formula. Moving boundaries may generate singularities when they touch. We establish two sets of sufficient conditions on the moving boundaries that guarantee that the variation of the local time of the associated reflected Brownian motion is, respectively, finite and infinite. We also apply these results to study the semimartingale property of a class of two-dimensional reflected Brownian motions.