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Probability that the maximum of the reflected Brownian motion over a finite interval \([0,t]\) is achieved by its last zero before \(t\). (English) Zbl 1329.60292
Summary: We calculate the probability \(p_c\) that the maximum of a reflected Brownian motion \(U\) is achieved on a complete excursion, i.e. \(p_c:=P\big(\overline{U}(t)=U^*(t)\big)\) where \(\overline{U}(t)\) (respectively \(U^*(t))\) is the maximum of the process \(U\) over the time interval \([0,t]\) (respectively \(\big[0,g(t)\big]\) where \(g(t)\) is the last zero of \(U\) before \(t\)).

MSC:
60J65 Brownian motion
60G70 Extreme value theory; extremal stochastic processes
60G17 Sample path properties
60J25 Continuous-time Markov processes on general state spaces
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