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The paper studies the question of whether the classical mirror andsynchronous couplings of two Brownian motions minimise and maximise,respectively, the coupling time of the corresponding geometric Brownianmotions. We establish a characterisation of the optimality of the two couplingsover any finite time horizon and show that, unlike in the case of Brownianmotion, the optimality fails in general even if the geometric Brownian motionsare martingales. On the other hand, we prove that in the cases of the ergodicaverage and the infinite time horizon criteria, the mirror coupling and thesynchronous coupling are always optimal for general (possibly non-martingale)geometric Brownian motions. We show that the two couplings are efficient if andonly if they are optimal over a finite time horizon and give a conjecturalanswer for the efficient couplings when they are suboptimal.

Mirror and Synchronous Couplings of Geometric Brownian Motions