I don't have a background in stochastic calculus.
Question: what is the autocorrelation of the Wiener process($W_{t}$)? $$\int_{0}^{T}W_{t}W_{t+\tau}dt=?$$
The facts I know so far are Itô's lemma and Itô isometry, but I do not know how to apply them. Also, I found a similar question.
Furthermore, I did a simulation and found out that the expectation of the integral satisfies $$\mathbb{E}\bigg{[}\int_{0}^{T}W_{t}W_{t+\tau}dt\bigg{]}=\frac{1}{2}(\tau-T)^{2}$$.
Can anyone help me figure out how to do it analytically? thank you!