I'm reading about stochastic differential equations. On page 4 of this mini course the stochastic differential equation is defined as follows:
Now return to the general case of equation $(1)$, write $\dfrac{d}{dt}$ instead of the dot: $$\dfrac{dX(t)}{dt} = b(X(t))dt + B(X(t))\dfrac{dW}{dt},$$ and finally multiply by $dt$:
$$(SDE)\,\,\,\begin{cases}dX(t) = b(X(t))dt + B(X(t))dW(t)\\ X(0) = x_0.\end{cases}$$
Question 1: Why does this define a differential equation? As far as I can tell there is no longer a derivative in this system.
On page 5 Ito's formula is introduced as follows:
Assume $n=1$ and $X(\cdot)$ solves the SDE $$dX = b(X)dt + dW$$ Suppose next that $u:\mathbb{R}\to\mathbb{R}$ is a given smooth function. We ask: what stochastic differential equation does $$Y(t) := u(X(t))$$ solve? Offhand, we would guess from $(3)$ that $$dY = u'dX = u'bdt + u'dW,$$ according to the chain rule, where $' = \dfrac{d}{dx}$. This is wrong however!
Question 2: Apparently it's wrong anyway, but how would you arrive at $dY = u'dX = u'bdt + u'dW$ at all? If $dX = b(X)dt + dW$ and $Y = u(X(t))$ then I would expect $dY = du(X(t))$.