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

Kozachenko, Yuriy; Vasylyk, Olga

Theory Stoch. Process. 12, No. 28, Part 3-4, 55-62 (2006).

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.

Reviewer: A. D. Borisenko (Kyïv)

MSC:

60G18	Self-similar stochastic processes
60G15	Gaussian processes

Keywords:

fractional Brownian motion; \(\varphi\)-sub-Gaussian processes; series expansions; simulation