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On the representation of fractional Brownian motion as an integral with respect to ${\left(dt\right)}^{a}$. (English) Zbl 1082.60029
Summary: Maruyama introduced the notation $db\left(t\right)=w\left(t\right){\left(dt\right)}^{1/2}$ where $w\left(t\right)$ is a zero-mean Gaussian white noise, in order to represent the Brownian motion $b\left(t\right)$. Here, we examine in which way this notation can be extended to Brownian motion of fractional order $a$ (different from $1/2\right)$ 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 $0◃a\le 1/2$, and processes with long-range dependence, with $1/2◃a\le 1$.

##### MSC:
 60G15 Gaussian processes