## On the representation of fractional Brownian motion as an integral with respect to $$(dt)^a$$.(English)Zbl 1082.60029

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

### MSC:

 60G15 Gaussian processes
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### References:

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