HAL : hal-00077096, version 1

DOI : 10.1016/j.anihpb.2006.09.007

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Annales de l'Institut Henri Poincare (B) Probability and Statistics 43, 6 (2007) 655-676

On maximum increase and decrease of Brownian motion

Paavo P. Salminen 1, Pierre P. Vallois 2, 3

(2007)

The joint distribution of maximum increase and decrease for Brownian motion up to an independent exponential time is computed. This is achieved by decomposing the Brownian path at the hitting times of the infimum and the supremum before the exponential time. It is seen that an important element in our formula is the distribution of the maximum decrease for the three dimensional Bessel process with drift started from 0 and stopped at the first hitting of a given level. From the joint distribution of the maximum increase and decrease it is possible to calculate the correlation coefficient between these at a fixed time and this is seen to be -0.47936... .

1 :	Mathematical department [Abo Finland] (MD)
	Abo Akademi university
2 :	Institut Elie Cartan Nancy (IECN)
	CNRS : UMR7502 – INRIA – Université Henri Poincaré - Nancy I – Université Nancy II – Institut National Polytechnique de Lorraine (INPL)
3 :	TOSCA (INRIA Sophia Antipolis / INRIA Lorraine / IECN)
	INRIA – CNRS : UMR7502 – Université Henri Poincaré - Nancy I – Université Nancy II – Institut National Polytechnique de Lorraine (INPL)

Domaine

:

Mathématiques/Mathématiques générales

Mots Clés : $h$-transform – time reversal – path decompositions – Brownian motion with drift – excursion process – maximum process – Itô measure – maximum drawdown – covariance – Catalan's constant

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Contributeur : Pierre Vallois <>

Soumis le : Lundi 29 Mai 2006, 14:39:31

Dernière modification le : Vendredi 29 Avril 2011, 09:48:39