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arXiv · 1902.06963

Covariance of the running range of a Brownian trajectory

Brandon Annesi·Enzo Marinari·Gleb Oshanin
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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 - 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.

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Brandon Annesi, Enzo Marinari, Gleb Oshanin. 2019-02-19. Covariance of the running range of a Brownian trajectory. https://doi.org/10.1088/1751-8121/ab306c

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