Take as an example the computation of one of the basic Ito integrals, namely $\int^T_0XdX=\frac{1}{2}(X^2(T) -T)$, where X is the usual Wiener process.
Do we have any heuristic explanation for the $-T$ term? Why the introduction of the stochastic evolution (i.e. the non-derivability of the function) lowers the result by a factor equal to the lenght of the interval?
The math is clear to me but I woiuld like to get an insight, if it exists.