Brownian meander factor for the spatial axis must be square root of the scaling factor for the time axis. The process resulting from this snip-and-scale procedure is a Brownian meander. As the name suggests, it is a piece of Brownian motion that spends all its time away from its starting point formula_5. The transition density formula_12 of Brownian meander is described as follows: For formula_13 and formula_14, and writing we have and In particular, i.e. formula_19 has the Rayleigh distribution with parameter 1, the same distribution as formula_20, where formula_21 is an exponential random variable with parameter 1. Brownian meander In the
Fractional Brownian motion stochastic integrals". In general though, unlike integrals with respect to regular Brownian motion, fractional stochastic integrals are not semimartingales. Just as Brownian motion can be viewed as white noise filtered by formula_8 (i.e. integrated), fractional Brownian motion is white noise filtered by formula_9 (corresponding to fractional integration). Practical computer realisations of an "fBm" can be generated, although they are only a finite approximation. The sample paths chosen can be thought of as showing discrete sampled points on an "fBm" process. Three realizations are shown below, each with 1000 points of an "fBm" with Hurst parameter 0.75. Realizations of three different