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Optimal coupling of jumpy Brownian motion on the circle

Part of: Stochastic systems and control Stochastic processes Markov processes

Published online by Cambridge University Press:  04 July 2023

Stephen B. Connor Open the ORCID record for Stephen B. Connor [Opens in a new window]  and
Roberta Merli
Stephen B. Connor*
Affiliation:
University of York
Roberta Merli*
Affiliation:
University of York
*
*Postal address: Department of Mathematics, University of York, York, YO10 5DD, UK.
*Postal address: Department of Mathematics, University of York, York, YO10 5DD, UK.
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Abstract

Consider a Brownian motion on the circumference of the unit circle, which jumps to the opposite point of the circumference at incident times of an independent Poisson process of rate $\lambda$. We examine the problem of coupling two copies of this ‘jumpy Brownian motion’ started from different locations, so as to optimise certain functions of the coupling time. We describe two intuitive co-adapted couplings (‘Mirror’ and ‘Synchronous’) which differ only when the two processes are directly opposite one another, and show that the question of which strategy is best depends upon the jump rate $\lambda$ in a non-trivial way. We also provide an explicit description of a (non-co-adapted) maximal coupling for any jump rate in the case that the two jumpy Brownian motions begin at antipodal points of the circle.

Keywords

Co-adapted couplingmirror and synchronous couplingmaximal couplingstochastic controlHamilton–Jacobi–Bellman equation

MSC classification

Primary: 93E20: Optimal stochastic control
Secondary: 60J65: Brownian motion 60G51: Processes with independent increments; L'evy processes
Type
Original Article
Information
Journal of Applied Probability , Volume 61 , Issue 1 , March 2024 , pp. 18 - 32
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

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References

Aldous, D. (1983). Random walks on finite groups and rapidly mixing Markov chains. In Séminaire de Probabilités XVII 1981/82, eds. J. Azéma and M. Yor. Springer, Berlin, pp. 243–297.CrossRefGoogle Scholar
Banerjee, S. and Kendall, W. S. (2017). Rigidity for Markovian maximal couplings of elliptic diffusions. Prob. Theory Relat. Fields 168, 55–112.CrossRefGoogle Scholar
Connor, S. B. (2013). Optimal coadapted coupling for a random walk on the hyper-complete graph. J. Appl. Prob. 50, 1117–1130.CrossRefGoogle Scholar
Connor, S. B. and Jacka, S. D. (2008). Optimal co-adapted coupling for the symmetric random walk on the hypercube. J. Appl. Prob. 45, 703–713.CrossRefGoogle Scholar
Connor, S. B. and Merli, R. (2022). Optimal coupling of jumpy Brownian motion on the circle. Preprint, arXiv:2203.14791.Google Scholar
Goldstein, S. (1979). Maximal coupling. Z. Wahrscheinlichkeitsth. 46, 193–204.CrossRefGoogle Scholar
Griffeath, D. (1975). A maximal coupling for Markov chains. Z. Wahrscheinlichkeitsth. 31, 95–106.CrossRefGoogle Scholar
Hsu, E. P. and Sturm, K. T. (2013). Maximal coupling of Euclidean Brownian motions. Commun. Math. Statist. 1, 93–104.CrossRefGoogle Scholar
Jacka, S. D., Mijatović, A. and Širaj, D. (2014). Mirror and synchronous couplings of geometric Brownian motions. Stoch. Process. Appl. 124, 1055–1069.CrossRefGoogle Scholar
Kendall, W. S. (2009). Brownian couplings, convexity, and shy-ness. Electron. Commun. Prob. 14, 66–80.CrossRefGoogle Scholar
Kendall, W. S. (2015). Coupling, local times, immersions. Bernoulli 21, 1014–1046.CrossRefGoogle Scholar
Kuwada, K. (2007). On uniqueness of maximal coupling for diffusion processes with a reflection. J. Theor. Prob. 20, 935–957.CrossRefGoogle Scholar
Lindvall, T. and Rogers, L. C. G. (1986). Coupling of multidimensional diffusions by reflection. Ann. Prob. 14, 860–872.CrossRefGoogle Scholar
Merli, R. (2021). Probabilistic coupling: Mixing times and optimality. PhD thesis, University of York.Google Scholar
Øksendal, B. K. and Sulem, A. (2007). Applied Stochastic Control of Jump Diffusions. Springer, Berlin.CrossRefGoogle Scholar
Pitman, J. W. (1976). On coupling of Markov chains. Z. Wahrscheinlichkeitsth. 35, 315–322.CrossRefGoogle Scholar
Rogers, L. C. G. and Williams, D. (2000). Diffusions, Markov Processes and Martingales, Vol. 2. Cambridge University Press.Google Scholar
Sverchkov, M. Y. and Smirnov, S. N. (1990). Maximal coupling of D-valued processes. Dokl. Math. 41, 352–354.Google Scholar