Hostname: page-component-8448b6f56d-c47g7 Total loading time: 0 Render date: 2024-04-16T09:13:59.960Z Has data issue: false hasContentIssue false
  • English
  • Français

Coupling the Kolmogorov diffusion: maximality and efficiency considerations

Part of: Markov processes Stochastic processes

Published online by Cambridge University Press:  25 July 2016

Sayan Banerjee  and
Wilfrid S. Kendall
Sayan Banerjee*
Affiliation:
University of Warwick
Wilfrid S. Kendall*
Affiliation:
University of Warwick
*
Department of Statistics, University of Warwick, Coventry CV4 7AL, UK. Email address: sayan.banerjee@warwick.ac.uk
Department of Statistics, University of Warwick, Coventry CV4 7AL, UK. Email address: w.s.kendall@warwick.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

This is a case study concerning the rate at which probabilistic coupling occurs for nilpotent diffusions. We focus on the simplest case of Kolmogorov diffusion (Brownian motion together with its time integral or, more generally, together with a finite number of iterated time integrals). We show that in this case there can be no Markovian maximal coupling. Indeed, there can be no efficient Markovian coupling strategy (efficient for all pairs of distinct starting values), where the notion of efficiency extends the terminology of Burdzy and Kendall (2000). Finally, at least in the classical case of a single time integral, it is not possible to choose a Markovian coupling that is optimal in the sense of simultaneously minimizing the probability of failing to couple by time t for all positive t. In recompense for all these negative results, we exhibit a simple efficient non-Markovian coupling strategy.

Keywords

Brownian motionBrownian time integralco-adapted couplingcouplingefficient couplingfiltrationfinite-look-ahead couplinghypoelliptic diffusionimmersed coupling; Karhunen‒Loève expansionKolmogorov diffusionMarkovian couplingmaximal couplingnilpotent diffusionoptimal Markovian couplingreflection couplingsynchronous coupling

MSC classification

Primary: 60G05: Foundations of stochastic processes
Secondary: 60J60: Diffusion processes
Type
Research Article
Information
Advances in Applied Probability , Volume 48 , Issue A: Probability, Analysis and Number Theory , July 2016 , pp. 15 - 35
Copyright
Copyright © Applied Probability Trust 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Alili, L. and Wu, C. T. (2014).Müntz linear transforms of Brownian motion.Electron. J. Prob. 19, 15pp.CrossRefGoogle Scholar
[2] Aue, A.,Horváth, L. and Hus̆ková, M. (2009).Extreme value theory for stochastic integrals of Legendre polynomials.J. Multivariate Anal. 100,1029–1043.CrossRefGoogle Scholar
[3] Bailleul, I. (2008).Poisson boundary of a relativistic diffusion.Prob. Theory Relat. Fields 141,283–329.CrossRefGoogle Scholar
[4] Banerjee, S. and Kendall, W. S. (2014).Rigidity for Markovian maximal couplings of elliptic diffusions. Preprint. Available at http://arxiv.org/abs/1412.2647v1.Google Scholar
[5] Ben Arous, G.,Cranston, M. and Kendall, W. S. (1995).Coupling constructions for hypoelliptic diffusions: two examples. In Stochastic Analysis (Ithaca, NY, 1993; Proc. Sympos. Pure Math. 57), eds M. C. Cranston and M. A. Pinsky,American Mathematical Society,Providence, RI, pp. 193–212.CrossRefGoogle Scholar
[6] Burdzy, K. and Kendall, W. S. (2000).Efficient Markovian couplings: examples and counterexamples.Ann. Appl. Prob. 10,362–409.CrossRefGoogle Scholar
[7] Da Prato, G. and Zabczyk, J. (2014).Stochastic Equations in Infinite Dimensions (Encyclopedia Math. Appl. 152),2nd edn.Cambridge University Press.CrossRefGoogle Scholar
[8] Doeblin, W. (1938).Exposé de la théorie des chaînes simples constantes de Markoff à un nombre fini d'états.Revue Math. de l'Union Interbalkanique 2,77–105.Google Scholar
[9] Goldstein, S. (1979).Maximal coupling.Prob. Theory Relat. Fields 46,193–204.Google Scholar
[10] Griffeath, D. (1975).A maximal coupling for Markov chains.Z. Wahrscheinlichkeitsth. 31,95–106.CrossRefGoogle Scholar
[11] Groeneboom, P.,Jongbloed, G. and Wellner, J. A. (1999).Integrated Brownian motion, conditioned to be positive.Ann. Prob. 27,1283–1303.CrossRefGoogle Scholar
[12] Jansons, K. M. and Metcalfe, P. D. (2007).Optimally coupling the Kolmogorov diffusion, and related optimal control problems.LMS J. Comput. Math. 10,1–20.CrossRefGoogle Scholar
[13] Kearney, M. J. and Majumdar, S. N. (2005).On the area under a continuous time Brownian motion till its first-passage time.J. Phys. A 38,4097–4104.CrossRefGoogle Scholar
[14] Kendall, W. S. (2007).Coupling all the Lévy stochastic areas of multidimensional Brownian motion.Ann. Prob. 35,935–953.CrossRefGoogle Scholar
[15] Kendall, W. S. (2010).Coupling time distribution asymptotics for some couplings of the Lévy stochastic area. In Probability and Mathematical Genetics: Papers in Honour of Sir John Kingman (London Math. Soc. Lecture Note Ser. 378), eds N. H. Bingham and C. M. Goldie,Cambridge University Press, pp. 446–463.CrossRefGoogle Scholar
[16] Kendall, W. S. (2015).Coupling, local times, immersions.Bernoulli 21,1014–1046.CrossRefGoogle Scholar
[17] Kendall, W. S. and Price, C. J. (2004).Coupling iterated Kolmogorov diffusions.Electron. J. Prob. 9,382–410.CrossRefGoogle Scholar
[18] Kunita, H. (1997).Stochastic Flows and Stochastic Differential Equations (Camb. Stud. Adv. Math. 24).Cambridge University Press.Google Scholar
[19] Kuwada, K. (2009).Characterization of maximal Markovian couplings for diffusion processes.Electron. J. Prob. 14,633–662.CrossRefGoogle Scholar
[20] McKean, H. P., Jr. (1963).A winding problem for a resonator driven by a white noise.J. Math. Kyoto Univ. 2,227–235.Google Scholar
[21] Pitman, J. W. (1976).On coupling of Markov chains.Z. Wahrscheinlichkeitsth. 35,315–322.CrossRefGoogle Scholar
[22] Rosenthal, J. S. (1997).Faithful couplings of Markov chains: now equals forever.Adv. Appl. Math. 18,372–381.CrossRefGoogle Scholar
[23] Smith, A. M. (2013).A Gibbs sampler on the n-simplex.Ann. Appl. Prob. 24,114–130.Google Scholar
[24] Sverchkov, M. Yu. and Smirnov, S. N. (1990).Maximal coupling for processes in D[0,∞].Dokl. Akad. Nauk SSSR 311,1059–1061 (in Russian). English translation:Soviet Math. Dokl. 41 (1990),352–354.Google Scholar
[25] Thorisson, H. (1994).Shift-coupling in continuous time.Prob. Theory Relat. Fields 99,477–483.CrossRefGoogle Scholar
[26] Wellner, J. A. and Smythe, R. T. (2002).Computing the covariance of two Brownian area integrals.Statist. Neerlandica 56,101–109.CrossRefGoogle Scholar
[27] Yan, L. (2004).Two inequalities for iterated stochastic integrals.Arch. Math. (Basel) 82,377–384.CrossRefGoogle Scholar