Probability that the maximum of the reflected Brownian motion over a finite interval [0; t] is achieved by its last zero before t

Agnès Lagnoux (1) , Sabine Mercier (1) , Pierre Vallois (2, 3)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 BIGS - Biology, genetics and statistics
3 IECL - Institut Élie Cartan de Lorraine

Probability that the maximum of the reflected Brownian motion over a finite interval [0, t] is achieved by its last zero before t Abstract We calculate the probability pc that the maximum of a reflected Brownian motion U is achieved on a complete excursion, i.e. pc := P U (t) = U * (t) where U (t) (respectively U * (t)) is the maximum of the process U over the time interval [0, t] (resp. 0, g(t) where g(t) is the last zero of U before t).

Mots clés: Reflected Brownian motion; Bessel process; local score; Brownian meander and Brownian bridge; Brownian excursions; Gamma function

Domaines: Statistiques [math.ST]

Agnès Lagnoux, Sabine Mercier, Pierre Vallois. Probability that the maximum of the reflected Brownian motion over a finite interval [0; t] is achieved by its last zero before t. Electronic Communications in Probability, 2015, ⟨10.1214/ECP.vVOL-PID⟩. ⟨hal-01214773⟩