Mirror and Synchronous Couplings of Geometric Brownian Motions
AbstractThe paper studies the question of whether the classical mirror and synchronous couplings of two Brownian motions minimise and maximise, respectively, the coupling time of the corresponding geometric Brownian motions. We establish a characterisation of the optimality of the two couplings over any finite time horizon and show that, unlike in the case of Brownian motion, the optimality fails in general even if the geometric Brownian motions are martingales. On the other hand, we prove that in the cases of the ergodic average and the infinite time horizon criteria, the mirror coupling and the synchronous coupling are always optimal for general (possibly non-martingale) geometric Brownian motions. We show that the two couplings are efficient if and only if they are optimal over a finite time horizon and give a conjectural answer for the efficient couplings when they are suboptimal.
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Bibliographic InfoPaper provided by arXiv.org in its series Papers with number 1304.1999.
Date of creation: Apr 2013
Date of revision: Oct 2013
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Web page: http://arxiv.org/
This paper has been announced in the following NEP Reports:
- NEP-ALL-2013-10-25 (All new papers)
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