Brownian motion itself can be described approximately by a Langevin equation, an equation which involves a random force field representing the effect of the thermal fluctuations of the solvent on the Brownian particle. On long timescales, the mathematical Brownian motion is well described by a Langevin equation. On small timescales, inertial effects are prevalent in the Langevin equation. However the mathematical "Brownian motion" is exempt of such inertial effects. Note that inertial effects have to be considered in the Langevin equation, otherwise the equation becomes singular. so that simply removing the inertia term from this equation would not yield an exact description,
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