Question: if $W(t)$ is a standard Brownian motion with $W(0)=0$, what is the linear coefficient between the stochastic processes $W(t)$ and $I(t)=\int_0^t W(s)ds$?
I argued as follows: what we want is the coefficient of the differential product $dW(t)dI(t)$. Then, write this as $dW(t)\cdot W(t)dt$. Now the presence of a product of the form $dW(t)dt$ which is equal to zero forces the prduct $dW(t)dI(t)=0$, which seems to imply that the correlation is zero. Is it this an acceptable argument? Or did I make a mistake??