Before revision:  A basic version of It\^o's lemma shows that our definition of the It\^o integral agrees with F\"ollmer's. Finally, we propose a tentative definition and show the existence of \int_0^t\phi d\omega for a c\`adl\`ag integrand \phi with a finite variation index and a c\`adl\`ag integrator \omega with jumps bounded in a predictable manner.
After revision:  For example, one of the results shows the existence of \int_0^t\phi d\omega for a c\`adl\`ag integrand \phi with a finite variation index and a c\`adl\`ag integrator \omega with jumps bounded in a predictable manner.