Wiener Integration with Respect to Fractional Brownian Motion

The Elements of Fractional Calculus

Let α > 0 (and in most cases below α < 1 though this is not obligatory). Define the Riemann–Liouville left- and right-sided fractional integrals on (a, b) of order α by

$$\left( {I_{a + }^\alpha f} \right)\left( x \right): = \frac{1}{{\Gamma \left( \alpha \right)}}\int\limits_a^x {f\left( t \right)\left( {x - t} \right)^{^{\alpha - 1} } dt,} $$

and

$$\left( {I_{b - }^\alpha f} \right)\left( x \right): = \frac{1}{{\Gamma \left( \alpha \right)}}\int\limits_x^b {f\left( t \right)\left( {t - x} \right)^{^{\alpha - 1} } dt,} $$

respectively.

We say that the function $f \in D\left( {I_{a + \left( {b - } \right)}^\alpha } \right)$ (the symbol $D\left( \cdot \right)$ denotes the domain of the corresponding operator), if the respective integrals converge for almost all (a.a.) $x \in \left( {a,b} \right)$ (with respect to (w.r.t.) Lebesgue measure).

The Riemann-Liouville fractional integrals on R are defined as

$$\left( {I_ + ^\alpha f} \right)\left( x \right): = \frac{1}{{\Gamma \left( \alpha \right)}}\int\limits_{ - \infty }^x {f\left( t \right)\left( {x - t} \right)^{\alpha - 1} } dt,$$

and

$$\left( {I_ - ^\alpha f} \right)\left( x \right): = \frac{1}{{\Gamma \left( \alpha \right)}}\int\limits_x^\infty {f\left( t \right)\left( {t - x} \right)^{\alpha - 1} } dt,$$

respectively.

The function $f \in D\left( {I_ \pm ^\alpha } \right)$ if the corresponding integrals converge for a.a.$x \in R$. According to (SKM93), we have inclusion $L_p \left( R \right) \subset D\left( {I_ \pm ^\alpha } \right),1 \le p < \frac{1}{\alpha }.$. Moreover, the following Hardy–Littlewood theorem holds.