Computing Covariance of Integrated Brownian Motions

Question

I have no idea on how to proceed finding:

$$cov[W(1),\int^1_0 rW(r)dr]$$

I can rewrite it as

$$cov[\int_0^1 dW(r),\int^1_0 rw(r)dr]$$

but it really leads me nowhere.

Any ideas, hints, suggestions?

Thanks!

Answer 1

Note that $E[W(1)]=0$ and $E[\int_{0}^{1}rW(r)dr]=\int_{0}^{1}E[rW(r)]dr=0$. By the definition of covariance, \begin{align*} \text{cov}[W(1),\int_{0}^{1}rW(r)dr]&=E[W(1)\int_{0}^{1}rW(r)dr]\\ &=E[\int_{0}^{1}rW(r)W(1)dr]&\text{$W(1)$ is independent of $r$}\\ &=\int_{0}^{1}E[rW(r)W(1)]dr\\ &=\int_{0}^{1}r^2dr&\text{$E[W(r)W(1)]=r\wedge1$}\\ &=\frac{1}{3}. \end{align*}