In stochastic calculus, what is the differentiation of $W(t)$?

Question

I am very confused dealing $dW(t)$, what is it? $W(t)$ is nowhere differentiable, we cannot write $W'(t)~dt$, but $dW(t)$ is a notation often used in my professor's notes.

Answer 1

Whenever you see an SDE of the form $$ dX_{s}=a(s,X_{s})ds+b(s,X_{s})dW(s), $$ just remember that this is simply a "short form" for $$ X_{s}=x+\int_{t}^{s}a(u,X_{u})du+\int_{t}^{s}b(u,X_{u})dW(u) $$ where $x$ is the initial condition at time $t$.

That is, the only place $dW(u)$ appears is in an Ito integral, whose definition you might already be familiar with. No derivatives are considered.

Addendum: While $dW(t)dW(t)=dt$ is just short form to describe the quadratic variation of the process $[W,W](t)=t$, I am not sure what the notation $dW(t_1)dW(t_2)$ could mean.