Consider the one-dimensional diffusion process $dX_t = \mu_tdt + \sigma_tdB_t$ and function $f : \mathbb{R} \to \mathbb{R}$, which is twice differentiable. Here we have another SDE by using Itô lemma as follows; $$ d f\left(t, X_t\right)=\left(\frac{\partial f}{\partial t}+\mu_t \frac{\partial f}{\partial x}+\frac{\sigma_t^2}{2} \frac{\partial^2 f}{\partial x^2}\right) d t+\sigma_t \frac{\partial f}{\partial x} d B_t $$
I know that Itô lemma is usually used to solve SDE about $X_t$. However, I also would like to know about this new SDE of $f(t, X_t)$.
Does this new SDE have any formal name? Are there any properties about this new SDE, for example the existence and uniqueness of the solution or the convergence to stationary distribution?
