Abstract
Maruyama introduced the notation d b ( t ) = w ( t ) ( d t ) 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 ⊲ a ≤ 1 / 2 , and processes with long-range dependence, with 1 / 2 ⊲ a ≤ 1 .
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