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?



Ito's lemma in differential form
I think there's an issue with definitions here. Ito's lemma in differential form only makes sense in the context of the integral form. It makes absolutely no sense to speak of $dW(s)/ds$, where $W$ is Brownian motion since it's nowhere differentiable. We DEFINE $dX\_t = sdB\_t+mdt$
by saying that this is shorthand for $X(t)-X(0)= \int\_0^t s dB\_s + \int\_0^t m ds$ . To reiterate, the differential form is a shorthand way of writing the integral form. It's like the notion of defining weak derivatives. You can't speak with a straight face about the derivative (in the usual sense) of say, a dirac delta function, but you can define it through an integral.


