Why isn't the Ito integral just the Riemann-Stieltjes integral?

Solution 1:

First of all, Brownian motion is almost surely nondifferentiable meaning that you can't just apply the rule you cite to Stieltjes integrals.

Second, in essence the Ito integral is Riemann Stieltjes integration when you observe the path of Brownian motion, but not entirely. You can think of it as Stieltjes integration where the "integrator" as you call it has an extra variable of dependence, in this case on the probability space of realizable Brownian paths. Specifically $B_t$ is a random variable. The definition is then essentially the same as its a limit over partitions. However Brownian motion is not of bounded variation so you cannot apply the usual definitions of Riemann Stieltjes to evaluate the integrals along some realized path. In particular you get the integral evaluated as a limit of sums over partitions converges in probability.

Solution 2:

You can't simply integrate with regards to Brownian motion pathwise because it's not of bounded variation (because such functions are differentiable almost everywhere) and functions of bounded variation are precisely those with respect to which you can compute the Riemann–Stieltjes integrals.