Hostname: page-component-76fb5796d-r6qrq Total loading time: 0 Render date: 2024-04-26T15:15:51.589Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Location of the Maximum of a Continuous Stochastic Process

Part of: Stochastic processes

Published online by Cambridge University Press:  30 January 2018

Leandro P. R. Pimentel
Leandro P. R. Pimentel*
Affiliation:
Federal University of Rio de Janeiro
*
∗ Postal address: Federal University of Rio de Janeiro, Postal Code 68530, 21.945-970, Rio de Janeiro, Brazil, Email address: leandro@im.ufrj.br
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this short article we will provide a sufficient and necessary condition to have uniqueness of the location of the maximum of a stochastic process over an interval. The result will also express the mean value of the location in terms of the derivative of the expectation of the maximum of a linear perturbation of the underlying process. As an application, we will consider a Brownian motion with variable drift. The ideas behind the method of proof will also be useful to study the location of the maximum, over the real line, of a two-sided Brownian motion minus a parabola and of a stationary process minus a parabola.

Keywords

Sample path propertiesmaximaargmaxBrownian motionstationary processparabolic drift

MSC classification

Primary: 60G17: Sample path properties
Secondary: 60G10: Stationary processes 60G15: Gaussian processes
Type
Research Article
Information
Journal of Applied Probability , Volume 51 , Issue 1 , March 2014 , pp. 152 - 161
Copyright
© Applied Probability Trust 

References

Balázs, M., Cator, E. and Seppäläinen, T. (2006). Cube root fluctuations for the corner growth model associated to the exclusion process. Electron. J. Prob. 11, 1094–1132.CrossRefGoogle Scholar
Cator, E. and Groeneboom, P. (2006). Second class particles and cube root asymptotics for Hammersley's process. Ann. Prob. 34, 1273–1295.Google Scholar
Corwin, I. and Hammond, A. (2014). Brownian Gibbs property for Airy line ensembles. Invent. Math. 195, 441–508.Google Scholar
Daniels, H. E. and Skyrme, T. H. R. (1985). The maximum of a random walk whose mean path has a maximum. Adv. Appl. Prob. 17, 85–99, 475.CrossRefGoogle Scholar
Ferrari, P. A. and Fontes, L. R. G. (1994). Current fluctuations for the asymmetric simple exclusion process. Ann. Prob. 22, 820–832.Google Scholar
Groeneboom, P. (1989). Brownian motion with a parabolic drift and Airy functions. Prob. Theory Relat. Fields 81, 79–109.Google Scholar
Groeneboom, P. (2011). Vertices of the least concave majorant of Brownian motion with parabolic drift. Electron. J. Prob. 16, 2234–2258.Google Scholar
Janson, S. (2013). Moments of the location of the maximum of Brownian motion with parabolic drift. Electron. Commun. Prob. 18, 8pp.CrossRefGoogle Scholar
Janson, S., Louchard, G. and Martin-Löf, A. (2010). The maximum of Brownian motion with parabolic drift. Electron. J. Prob. 15, 1893–1929.CrossRefGoogle Scholar
Johansson, K. (2003). Discrete polynuclear growth and determinantal processes. Commun. Math. Phys. 242, 277–329.Google Scholar
Kim, J. and Pollard, D. (1990). Cube root asymptotics. Ann. Statist. 18, 191–219.Google Scholar
Moreno Flores, G., Quastel, J. and Remenik, D. (2013). Endpoint distribution of directed polymers in 1+1 dimensions. Commun. Math. Phys. 317, 363–380.Google Scholar
Prähofer, M. and Spohn, H. (2002). Scale invariance of the PNG droplet and the Airy process. J. Statist. Phys. 108, 1071–1106.Google Scholar
Seppäläinen, T. (2012). Scaling for a one-dimensional directed polymer with boundary conditions. Ann. Prob. 40, 19–73.CrossRefGoogle Scholar