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0 vote down The differential form is just a suggestive shorthand for the integral form. It's just as valid as the integral form, because it is the same thing. Do you have any reference which suggests otherwise?
I have to correct your comment "But the numerator is zero whenever f(S)<=S(T)". The numerator is an expectation, so just a fixed number. It is not zero as long as there is a positive probability that f(S)>S(T).
