All the resources I have found state that a Stochastic Differential Equation has the form: $$dX_t=\mu(X_t,t)dt+\sigma(X_t,t)dB_t$$ Where $B_t$ is Brownian motion. My question has two parts. The first one is: why is an SDE given in this form when regular ODEs are given the apparently more general form $$\frac{dX}{dt}=f(t,X)?$$ By taking $X$ to be a vector and renaming derivatives to coordinates this expression captures everything from higher order to systems of DEs. Analogously, I would have expected a general SDE to instead look like $$dX_t=f(t,X_t,B_t)dt\tag{1}$$ But the canonical formulation seems to be just linear in the stochastic term. My guess here is that we can "augment" the state space by letting $Y_t=\begin{bmatrix} X_t \\B_t\end{bmatrix}$ and rewriting $(1)$ as $$dY_t=\begin{bmatrix}f(t,Y_t)\\ 0\end{bmatrix}dt+\begin{bmatrix} 0 \\ 1\end{bmatrix}dB_t.$$ Is this manipulation correct?
My second question is why can we only use Brownian motion (or perhaps more generally semimartingales) as integrators? Usually SDEs are introduced by discretizing a differential equation and summing the changes in the random process to arrive at a stochastic integral in the limit. Why can we not just replace $B_t$ by any stochastic process?