# Séminaire du LPTMS: Tridib Sadhu

## Generalized arcsine law in fractional Brownian motion

### Tridib Sadhu (LPT-ENS, Paris)

The three arcsine laws for the standard Brownian motion are a cornerstone of extreme value statistics. For a standard Brownian motion evolving in a time window, one can consider the following three observables: (1) the fraction of time it remained positive, (2) the last time it crossed the origin, (3) and the time when it reached its maximum. All three observables have the same cumulative probability distribution expressed as an arcsine function. I shall discuss how these three laws change for a fractional Brownian motion. The fractional Brownian motion is a non-Markovian Gaussian process indexed by Hurst exponent H which generalizes Brownian motion (H=1/2). I shall show that the three observables have different distributions for general H. I shall present a perturbation expansion scheme using which one can derive these probability distributions

**Date/Time : ** 09/05/2017 - *11:00 - 12:00*

**Location : ** LPTMS, salle 201, 2ème étage, Bât 100, Campus d'Orsay

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