On the local time of the Brownian motion
Let \(W(t)\), \(t \in [0,1]\), be a reflected Brownian motion, started at zero, and denote by \(\tau(\alpha)\) its local time at level \(\alpha \geq 0\). An explicit formula is derived for the density and the moments of the distribution of the random variable \(\tau(\alpha)\). The proof of these formulas uses a simple random walk approximation for the Brownian motion. One can compute explicitly the binomial moments for the local times of the simple random walk, and obtain the binomial moments for the limit distribution. From this one can compute the moments, and then the Laplace transform, and then by inversion, the distribution function and the density of the distribution.
scientific article; zbMATH DE number 854863