The traditional FTC says $$ \int_a^b f'(x) dx = f(b) - f(a)$$ if $f'$ is Riemann integrable.
I know that Ito's lemma is like a stochastic version of the FTC. But that is phrased in terms of the derivatives of a function of a stochastic process.
What about the stochastic analogue of the derivative itself? That is, the semigroup operator of a stochastic process $X_t$ is given by $$ Af(x) = \lim_{t \rightarrow 0} \frac{ \mathbb{E} [f(X_t) | X_0 = x] - f(x) }{t} $$ which I think of as an infinitesimal "expected rate of change" for a function $f$ of the stochastic process. Here $f$ usually has to be continuous and bounded, I think.
Can we say something like $$ \int_a^b Af(X_t) dX_t = f(b)-f(a) ?$$ I've seen a FTC for Ito's lemma but not for semigroups.
