On the representation of fractional Brownian motion as an integral with respect to (dt)a

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⊲a≤1/2, and processes with long-range dependence, with 1/2⊲a≤1.