Expectation of a stochastic differential

Question

Given

\begin{equation} df(X_{t})=\left(\mu_{t}{\frac {\partial f}{\partial x}}+{\frac {\sigma_{t}^{2}}{2}}{\frac {\partial^{2}f}{\partial x^{2}}}\right)dt+\sigma_{t}{\frac {\partial f}{\partial x}} dB_{t} \end{equation}

Is there some hope to make sense of $$ \mathbb{E}(df),\qquad \mathbb{E}\left(\frac{df}{dt}\right)\overset{?}{=}\frac{d}{dt}\mathbb{E}(f)$$

Answer 1

The writing $df$ is just a shorthand. The proper and meaningful way to write it is in integral form

$$ f(X_t) = f(X_0) + \int_{0}^{t} \bigg( \mu_s \partial_x f + \frac{\sigma_{s}^{2}}{2} \partial_{xx}f \bigg) ds + \int_{0}^{t} \sigma_{s}^{2} \partial_{x}f dB_s $$

Now you may be able to compute the conditional expectation and from that the unconditional one. One usegul fact is that if the second integral is a martingal that expectation is null and you just have to compute the mean of the first part.

For the derivative I'd say we enter the world of Malliavin Calculus which I'm not too familiar with to be of help.