Brownian excursion the Airy function and formula_22 is the confluent hypergeometric function. Janson and Louchard (2007) show that and They also give higher-order expansions in both cases. Janson (2007) gives moments of formula_19 and many other area functionals. In particular, Brownian excursions also arise in connection with queuing problems, railway traffic, and the heights of random rooted binary trees. Brownian excursion In probability theory a Brownian excursion process is a stochastic process that is closely related to a Wiener process (or Brownian motion). Realisations of Brownian excursion processes are essentially just realizations of a Wiener process selected to satisfy certain conditions. In
Brownian motion was able to rule out that the motion was life-related, although its origin was yet to be explained. The first person to describe the mathematics behind Brownian motion was Thorvald N. Thiele in a paper on the method of least squares published in 1880. This was followed independently by Louis Bachelier in 1900 in his PhD thesis "The theory of speculation", in which he presented a stochastic analysis of the stock and option markets. The Brownian motion model of the stock market is often cited, but Benoit Mandelbrot rejected its applicability to stock price movements in part because these are