I have a continuous random variable $V_t$ for which I was able to show that
$$\mathbb{E}_t\left[\frac{dV}{dt}\right]=X_t.$$
I now want to write (in short-form notation) $dV_t=X_tdt$. How could I justify this writing? I was feeling pushed towards the Radon Nikodym derivative however I had difficulty with a formal justification. Thank you for thinking about it!