The definition of fractional order integral is well-known. Is there any definition for fractional order Riemann Stieltjes integral?
2026-03-26 05:54:40.1774504480
Fractional order Riemann Stieltjes integral
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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