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
Fractional Brownian motion Suppose we want to simulate the values of the "fBM" at times formula_12 using the Cholesky decomposition method. In order to compute formula_15, we can use for instance the Cholesky decomposition method. An alternative method uses the eigenvalues of formula_16: Note that the result is real-valued because formula_36. Note that since the eigenvectors are linearly independent, the matrix formula_39 is invertible. It is also known that where "B" is a standard Brownian motion and Where formula_47 is the Euler hypergeometric integral. Say we want to simulate an "fBm" at points formula_48. The integral may be efficiently computed by Gaussian quadrature.