Basically you'll find two versions of ito's lemma in the literature: an integral and a differential form. The integral form is based on an Riemann-Stieltjes-integral approach, the differential form is said to be the chain rule for stochastic processes.
My questions: Some purists tell you that only the integral form is valid and the differential form is a shortcut to that at most. Why do they think so? Others tell you that both forms are ok and are just two sides of the same coin. What is true now and why?