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

Keywords

Gaussian Process Fractional Derivative Wiener Process Fractional Brownian Motion Maximal Inequality 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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