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

Maruyama introduced the notation db(t)=w(t)(dt)1/2 where w(t)w(t) is a zero-mean Gaussian white noise, in order to represent the Brownian motion b(t)b(t). Here, we examine in which way this notation can be extended to Brownian motion of fractional order aa (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≤1/20⊲a≤1/2, and processes with long-range dependence, with 1/2⊲a≤11/2⊲a≤1.