A strong uniform approximation of fractional Brownian motion by means of transport processes
This paper contributes to simulation methods for fractional Brownian motion (fBm). For a Hurst index \(H\not=\frac12\) the authors use the Mandelbrot-van Ness integral representation of fBm \[ W_t=C_H\left\{\int_{-\infty}^0(t-s)^{H-\frac12}-(-s)^{H-\frac12}\,dB_s+\int_0^t(t-s)^{H-\frac12}\,dB_s\right\}, \] where \((B_s)_{s\in\mathbb R}\) is Brownian motion (Bm) on the real line, together with a strong approximation of Bm by transport processes which holds uniformly on bounded time-intervals. In fact the time-interval \((-\infty,t]\) is decomposed into \((-\infty,a]\), \([a,0]\), \([0,t]\) for \(a < 0\), where for the first integral Bm is replaced by its time reversed version on \([\frac1a,0)\). The approximating transport processes have paths whose absolute velocity \(v\) is constant and changes its sign at random times with inter-arrivals exponentially distributed of parameter \(v^2\). This results in strong uniform convergence theorems for fBm with Hurst index \(0 <H < \frac12\), respectively \(\frac12 < H <1\). A similar result has been obtained by \textit{T. Szabados} [Stochastic Processes Appl. 92, No. 1, 31--60 (2001; Zbl 1047.60032)] for \(\frac14 < H <1\) using a different strong approximation technique for fBm, where the rate of convergence in the paper under consideration is better in case \(H <\frac12(2-\log 2)\).
scientific article; zbMATH DE number 5614623
scientific article