In this paper the $L^1$-stochastic integral and the mixed stochastic integral of a process $Y$ with respect to a process $X$ is defined in a way that extends Riemann-Stieltjes integration of deterministic functions with respect to $X$. The $L^1$-integral will include the classical Ito integral. However, the concepts of "filtration" and adaptability do not play any role; instead, the $p$-variation of Dolean functions of the processes $X$ and $Y$ is the determining factor.
@article{1176988282,
author = {Towghi, Nasser},
title = {Stochastic Integration of Processes with Finite Generalized Variations. I},
journal = {Ann. Probab.},
volume = {23},
number = {3},
year = {1995},
pages = { 629-667},
language = {en},
url = {http://dml.mathdoc.fr/item/1176988282}
}
Towghi, Nasser. Stochastic Integration of Processes with Finite Generalized Variations. I. Ann. Probab., Tome 23 (1995) no. 3, pp. 629-667. http://gdmltest.u-ga.fr/item/1176988282/