I am looking to prove the following equation: $$\left(\int_s^t \sigma(X_r)dW_r\right)\left(\int_s^t \sigma(X_r)dW_r\right)^\top = \int_s^t \sigma(X_r)\sigma(X_r)^\top dr$$
for a function $\sigma(x)\in R^{d\times d}$ that is continuous and $W$ a d-dimensional Wiener process. I am not even sure if this is true, but it would suffice if these two terms were relatively close to each other. Since in the case that $d=1$, this is just the Itô-formula, I suspect this to be true.
I tried applying the multi-dimensional Itô-formula but this did not yield anything useful.
I am thankful for any hints.