  This paper describes two explicit couplings of standard Brownian motions $B$
and $V$, which naturally extend the mirror coupling and the synchronous
coupling and respectively maximise and minimise (uniformly over all time
horizons) the coupling time and the tracking error of two regime-switching
martingales. The generalised mirror coupling minimizes the coupling time of the
two martingales while simultaneously maximising the tracking error for all time
horizons. The generalised synchronous coupling maximises the coupling time and
minimises the tracking error over all co-adapted couplings. The proofs are
based on the Bellman principle. We give counterexamples to the conjectured
optimality of the two couplings amongst a wider classes of stochastic
integrals.
