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Zbl 1082.60029
Jumarie, Guy
On the representation of fractional Brownian motion as an integral with respect to $(dt)^a$. (English)
[J] Appl. Math. Lett. 18, No. 7, 739-748 (2005). ISSN 0893-9659

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 $0 \triangleleft a\le 1/2$, and processes with long-range dependence, with $1/2\triangleleft a\le 1$.

MSC 2000:: *60G15 Gaussian processes

Keywords: Maruama notation; stochastic differential equation; stochastic calculus of fractional order; Taylor's series of fractional order

Cited in: Zbl 1137.65001

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