I've read in a paper that, if $f$ is continuous, then
$$ \mathbb{d}\,\mathbb{E}(f(X_t)) =\mathbb{E}(\mathbb{d}f(X_t)) $$
where $X_t$ is a stochastic process and $\mathbb{d}$ is a differentiation, as in Itô's lemma. That a result like this holds makes sense, but is it obvious?
I've had a look at Oksendal's book, but I haven't found the result. Maybe I haven't looked enough, but any references you could provide will be appreciated.
In stochastic analysis, equations such as $df(X_t) = \dots$ are defined through integral equations (because, in general, there is no hope to define a derivative in the classical sense). For the identity
$$d\mathbb{E}(f(X_t)) = \mathbb{E}(d(f(X_t))$$
the best interpretation as an integral equation I can think of is
$$\int_0^s \mathbb{E}(f(X_s)) \, ds = \mathbb{E} \left( \int_0^s f(X_t) \, dt \right), \qquad s \geq 0.$$
This identity is a direct consequence of Fubini's theorem (whenever $\mathbb{E}(\int_0^s |f(X_t)| \, dt)<\infty$ and everything is nicely measurable).