## Simulation of fractional Brownian motion with given reliability and accuracy in $$C([0,1])$$.(English)Zbl 1141.60019

The authors deal with simulation of fractional Brownian motion defined on the interval $$[0,1]$$ with given reliability and accuracy in the space $$C([0,1])$$. Centered second order $$\text{Sub}_\varphi(\Omega)$$-processes defined on the interval $$[0,1]$$ with covariance function $$R(t,s)= {1\over2}\left(t^{2H}+s^{2H}-| t-s|^{2H}\right)$$ are considered. The parameter $$H$$ takes values in the interval $$(0,1)$$. In order to construct a model of such process the series expansion $Z_{t}= \sum_{n=1}^{\infty}{\sin( x_{nt})\over x_{n}}X_{n}+\sum_{n=1}^{\infty}{1-\cos( y_{nt})\over y_{n}}Y_{n}, \quad t\in[0,1]$ for $$\varphi$$-sub-Gaussian random processes with covariance function $$R$$ is used. Here $$X_{n}$$’s and $$Y_ n$$’s are independent random variables from the space $$\text{Sub}_\varphi(\Omega)$$. The $$x_ n$$’s are positive real zeros of the Bessel function $$J_{-H}$$ of the first kind and the $$y_ n$$’s are positive real zeros of the Bessel function $$J_{1-H}$$. Processes of fractional Brownian motion belong to the space $$\text{Sub}_\varphi(\Omega)$$ with $$\varphi(x) = x^2/2$$. Some examples of simulation of fractional Brownian motion with different values of parameter $$H$$ are presented.

### MSC:

 60G18 Self-similar stochastic processes 60G15 Gaussian processes