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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
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Abstract

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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 

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