An SDE of the form
$$dX_t = \mu(X_t,t)dt + \sigma(X_t,t)dB_t$$
is really short-hand notation for an equation involving Ito integrals:
$$X_t = X_0 + \int_0^t \mu(X_s,s)\,ds + \int_0^t \sigma(X_s,s)\,dB_s$$
Every book I have seen is stating that.
But that situation is unsettling to me. The parts of the SDE, i.e. $dX_t, \mu(X_t,t)dt$ and $\sigma(X_t,t)dB_t$ have no meaning on their one, although the notation suggests that they should have meaning. So
Is there a useful "differential" operator $d$ that can be applied to stochastic processes and gives meaning to the above SDE?
If the answer is in a sense negative, some sort of explanation on why that is would be appreciated.