If $X$ is a bounded left-continuous and piecewise constant process and if $Z$ is an arbitrary process, both adapted, then the stochastic integral $\int X dZ$ is defined as usual so as to conform with the sure case. In order to obtain a reasonable theory one needs to put a restriction on the integrator $Z$. A very modest one suffices; to wit, that $\int X_n dZ$ converge to zero in measure when the $X_n$ converge uniformly or decrease pointwise to zero. Daniell's method then furnishes a stochastic integration theory that yields the usual results, including Ito's formula, local time, martingale inequalities, and solutions to stochastic differential equations. Although a reasonable stochastic integrator $Z$ turns out to be a semimartingale, many of the arguments need no splitting and so save labor. The methods used yield algorithms for the pathwise computation of a large class of stochastic integrals and of solutions to stochastic differential equations.
"Stochastic Integration and $L^p$-Theory of Semimartingales." Ann. Probab. 9 (1) 49 - 89, February, 1981. https://doi.org/10.1214/aop/1176994509