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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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