I am considering two following problems. They involve two dependent Wiener processes: $W_{1t}$ and $W_{2t}$.
a) How to calculate the following (function $f$ is the same in both integrals)? Is there a specific Ito Isometry? $$ \mathbb{E}\Big(\int_0^Tf(W_{1t})dW_{2t} \hspace{1mm} \int_0^Tf(W_{2t})dW_{1t}\Big) $$
b) Is this quantity the same as in a)? $$ \mathbb{E}\Big[\Big(\int_0^Tf(W_{1t})dW_{2t}\Big)^2\Big] $$
Regarding a), I know the paper by Guillaume (2017), where he derives the following result: $$ \mathbb{E}\Big(\int_0^Tf(W_{1t})dW_{1t} \hspace{1mm} \int_0^Tf(W_{2t})dW_{2t}\Big)=\rho\int_0^T\mathbb{E}[f(W_{1t})f(W_{2t})]dt $$ where $\rho$ is the correlation coefficient between two Wiener processes. However, I am not clear how to deal with the case involving these cross-overs.
Just to provide the context I am working with, I am considering bivariate vector autoregressions involving two highly persistent autoregressive processes with the autoregressive coefficient $1+\frac{c}{T}$ ($c<0$ and $T$ is my sample size). I estimate the system using least squares and do the asymptotic analysis invoking Functional Central Limit Theorem. Therefore, we can say that this $f(W_t)$ is the noise component of Ornstein-Uhlenbeck process (Phillips, 1987): $\int_0^re^{(r-s)c}dW_s$. Since the system has process 1 and 2, we receive integrals of the form in a) and b).