# MEXIT: Maximal un-coupling times for stochastic processes

## TL;DR

This paper introduces MEXIT, a novel coupling framework that maximizes the duration two stochastic processes remain equal, contrasting traditional methods that minimize this time, with explicit constructions for discrete and continuous processes.

## Contribution

The paper develops the first explicit MEXIT construction for discrete-time countable state-space processes and extends it to continuous-time processes, including Brownian motions with different drifts.

## Key findings

- Explicit MEXIT construction for discrete-time processes.
- Extension of MEXIT to continuous-time processes.
- Application to Brownian motions with different drifts.

## Abstract

Classical coupling constructions arrange for copies of the \emph{same} Markov process started at two \emph{different} initial states to become equal as soon as possible. In this paper, we consider an alternative coupling framework in which one seeks to arrange for two \emph{different} Markov (or other stochastic) processes to remain equal for as long as possible, when started in the \emph{same} state. We refer to this "un-coupling" or "maximal agreement" construction as \emph{MEXIT}, standing for "maximal exit". After highlighting the importance of un-coupling arguments in a few key statistical and probabilistic settings, we develop an explicit \MEXIT construction for stochastic processes in discrete time with countable state-space. This construction is generalized to random processes on general state-space running in continuous time, and then exemplified by discussion of \MEXIT for Brownian motions with two different constant drifts.