MEXIT : Maximal uncoupling times for Markov processes

Classical coupling constructions arrange for copies of the same Markov process started at two dif- ferent 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 di erent Markov (or other stochastic) processes to remain equal for as long as possible, when started in the same state. We refer to this \un-coupling" or \maximal agreement" construction as 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 exempli ed by discussion of MEXIT for Brownian motions with two di erent constant drifts.