Wiener Integration with Respect to Fractional Brownian Motion

Part of the Lecture Notes in Mathematics book series (LNM, volume 1929)

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

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

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.


Gaussian Process Fractional Derivative Wiener Process Fractional Brownian Motion Maximal Inequality 
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