TITLE: Fractional order Riemann Stieltjes integral
QUESTION [9 upvotes]: The definition of fractional order integral is well-known. Is there any definition for fractional order Riemann Stieltjes integral?

REPLY [2 votes]: I'm not sure if this helps, but we do have the Riemann-Stieltjes integral with respect to fractional Brownian motion $B^H$ (a generalisation of Brownian motion). Let the FBM be defined as a Gaussian process with covariance function
\begin{equation*}
\frac{1}{2}(|t|^{2H}+|s|^{2H}-|t-s|^{2H})
\end{equation*}
where $H\in (0,1)$ is the Hurst index. Then we have the Riemann-Stieltjes integral representation
\begin{equation*}
Y=C+\int^T_0 H^Y_sdB^H_s,~T>0
\end{equation*}
where $Y$ is square integrable fractional Brownian functional. 
Esko Valkeila has done some interesting work on this and its relation to mathematical finance:

Riemann-Stieltjes Integrals With Respect to Fractional Brownian Motion and Applications http://lib.tkk.fi/Diss/2010/isbn9789526033389/isbn9789526033389.pdf
Riemann-Stieltles Integrals and Fractional Brownian Motion http://www.ambitprocesses.au.dk/fileadmin/pdfs/ambit/Ambit-10-valkeila.pdf