Mirror and synchronous couplings of geometric Brownian motions

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:124

Article

Jacka, Saul D.¹  Mijatovic, Aleksandar²  Siraj, Dejan¹ 

[1] Univ Warwick, Dept Stat, Coventry CV4 7AL, W Midlands, England

[2] Univ London Imperial Coll Sci Technol & Med, Dept Math, London SW7 2AZ, England

关键词: Mirror and synchronous coupling;    Coupling time;    Geometric Brownian motions;    Efficient coupling;    Optimal coupling;    Bellman's principle;   

DOI  :  10.1016/j.spa.2013.10.003

来源: Elsevier

【 摘 要 】

The 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. (C) 2013 Elsevier B.V. All rights reserved.

【 授权许可】

Free