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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
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