A few days ago, I disccussed my Phd thesis (thesis defense), one of the mathematical mistakes that the committe members alerted me to it, is how to write correctley an ordinary differential equation or a stochastic differential equation, they said to me that the correct way to write an ordinary differential equation is:
$\frac{dx(t)}{dt}=f(x,t)$ and $dx(t)=f(x,t)dt$ is false, and for a stochastic differential equation the right way is $dx(t)=f(x,t)dt+LdB(t)$, $L$: is square root of the gain matrix of the process noise.
what is the rationale behind these two expression, because for example I saw in many times that the two expression are used for the deterministic models?