On the representation of fractional Brownian motion as an integral with respect to \((dt)^a\). (English) Zbl 1082.60029

Summary: Maruyama introduced the notation \(db(t)=w(t)(dt)^{1/2}\) where \(w(t)\) is a zero-mean Gaussian white noise, in order to represent the Brownian motion \(b(t)\). Here, we examine in which way this notation can be extended to Brownian motion of fractional order \(a\) (different from \(1/2)\) defined as the Riemann-Liouville derivative of the Gaussian white noise. The rationale is mainly based upon the Taylor’s series of fractional order, and two cases have to be considered: processes with short-range dependence, that is to say with \(0 \triangleleft a\leq 1/2\), and processes with long-range dependence, with \(1/2\triangleleft a\leq 1\).


60G15 Gaussian processes
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