Covariance of Wiener Processes on the same Brownian Motion

Question

I am trying to solve $Cov(Tw_T,\int^{T}_{0}tdw_t)=\mathbb{E}[Tw_T\times\int^{T}_{0}tdw_t]$, my attempt is as below: \begin{split} \mathbb{E}[Tw_T\times\int^{T}_{0}tdw_t] & =T\mathbb{E}[\int^{T}_{0}dw_t\times\int^{T}_{0}tdw_t]\\ &=T\mathbb{E}[\int^{T}_{0}tdw_tdw_t]\\ &=T\mathbb{E}[\int^{T}_{0}tdt]\\ &=\frac{T^3}{2} \end{split} $\frac{T^3}{2}$ is the true answer. I think of the integrals as summation of brownian motion increments and therefore i found it intuitive to merge the integrals together (2nd equality). However I have never seen people solving like this (at least in the textbook) therefore I wonder if it's an appropriate way, otherwise how to solve? When can we merge integrals?

Answer 1

Most textbooks I know explain the reasons why $$E\left(\int_0^tu(s)\mathrm dw_s\cdot\int_0^tv(s)\mathrm dw_s\right)=\int_0^tu(s)v(s)\mathrm ds,$$ for every regular deterministic $u$ and $v$ and how to generalize this to more general settings.