On maximum increase and decrease of Brownian motion

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

1 MD - Mathematical department [Abo Finland]
2 IECN - Institut Élie Cartan de Nancy
3 TOSCA - Simuler et calibrer des modèles stochastiques

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... .

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

Domaines: Mathématiques générales [math.GM]

Paavo P. Salminen, Pierre P. Vallois. On maximum increase and decrease of Brownian motion. Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, 2007, 43 (6), pp.655-676. ⟨10.1016/j.anihpb.2006.09.007⟩. ⟨hal-00077096⟩