Suppose that $\sigma(t,T)$ is a deterministic process, where $t$ varies and $T$ is a constant. We also have that $t \in [0,T]$. Also $W(t)$ is a Wiener process.
My First Question
What is $\displaystyle \ \ d\int_0^t \sigma(u,T)dW(u)$? My lecture slides assert that it's equal to $\sigma(t,T)dW(t)$ but I'm not sure why. So my question is "Why"?
My Second Question
What is $\displaystyle \ \ d\int_a^t \sigma(u,T)dW(u)$, where $a \in (0,t)$.
I would appreciate careful explanation as I am not a mathematician. This is not homework, but I've put the tag on because it's on that level. Thank $\mathbf{YOU}$!