Why isn't the Ito integral just the Riemann-Stieltjes integral?
What I mean is, given a continuous function $f$, some path of standard brownian motion $B$, and the integral:
So what if we can't apply the change of variables formula to make sense of
the Riemann-Stieltjes integral never required differentiability of the integrator anyways.
Is there a reason to distinguish the Ito integral from the Riemann-Stieltjes integral above and beyond the need to develop a theory (Ito Calculus) to get around all the problems caused by the failure of change of variables?