The question how the extremal values of a stochastic process achieved on different time intervals are correlated to each other has been discussed within the last few years on examples of the running maximum of a Brownian motion, of a Brownian bridge and of a Slepian process. Here, we focus on the two-Time correlations of the running range of Brownian motion (BM)-the maximal extent of a Brownian trajectory on a finite time interval. We calculate exactly the covariance function of the running range and analyse its asymptotic behaviour. Our analysis reveals non-Trivial correlations between the value of the largest descent (rise) of a BM from the top to a bottom on some time interval, and the value of this property on a larger time interval.

Covariance of the running range of a Brownian trajectory / Annesi, B.; Marinari, E.; Oshanin, G.. - In: JOURNAL OF PHYSICS. A, MATHEMATICAL AND THEORETICAL. - ISSN 1751-8113. - 52:34(2019). [10.1088/1751-8121/ab306c]

Covariance of the running range of a Brownian trajectory

Marinari E.;
2019

Abstract

The question how the extremal values of a stochastic process achieved on different time intervals are correlated to each other has been discussed within the last few years on examples of the running maximum of a Brownian motion, of a Brownian bridge and of a Slepian process. Here, we focus on the two-Time correlations of the running range of Brownian motion (BM)-the maximal extent of a Brownian trajectory on a finite time interval. We calculate exactly the covariance function of the running range and analyse its asymptotic behaviour. Our analysis reveals non-Trivial correlations between the value of the largest descent (rise) of a BM from the top to a bottom on some time interval, and the value of this property on a larger time interval.
2019
extremal values of Brownian motion; running range; temporal correlations
01 Pubblicazione su rivista::01a Articolo in rivista
Covariance of the running range of a Brownian trajectory / Annesi, B.; Marinari, E.; Oshanin, G.. - In: JOURNAL OF PHYSICS. A, MATHEMATICAL AND THEORETICAL. - ISSN 1751-8113. - 52:34(2019). [10.1088/1751-8121/ab306c]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/1339811
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