# zbMATH — the first resource for mathematics

Fractional Brownian motion and the Markov property. (English) Zbl 0921.60067
Fractional Brownian motion is used as a model of signal or noise in various domains: geophysical data, communication processes, etc. It belongs to a class of long memory Gaussian processes that can be represented as linear functionals of an infinite-dimensional Markov process. Increments of this process depend on Hurst parameter $$H\in (0,1)$$: they are independent if and only if $$H=1/2$$ (Brownian motion), they are negative correlated if $$H<1/2$$, and if $$H>1/2$$ they are positive correlated. The authors give an efficient algorithm to approximate this process. Also, an ergodic theorem which applies to functionals of the type $$\int_{0}^{t}\varphi(V_{h}(s))ds$$, where $$V_{h}(s)=\int_{0}^{s}h(s-u)dB_{u}$$ is the finite long memory part with $$B$$ being a real Brownian motion, is given.

##### MSC:
 60J25 Continuous-time Markov processes on general state spaces 60G15 Gaussian processes 65C99 Probabilistic methods, stochastic differential equations 60F17 Functional limit theorems; invariance principles
Full Text: