How can one look at the evolution of covariance matrix using Ito lemma for a non-linear stochastic differential equation? when the stochastic term is a Brownian motion? $$ d\mathbf{X(t)} = \boldsymbol{F(X,t)}\, dt + \mathbf{G(X,t)}\, d\mathbf{B} $$
$$ \frac{d}{dt}Cov(X, X) = ? $$ when $Cov(X, X) = \langle X^2 \rangle - \langle X \rangle^2$.
We apply Itô lemma to $Y_{t}=(X_{t}-\mu_{t})^{2}$
$$dY_{t}=2(X_{t}-\mu_{t})dX_{t}-2(X_{t}-\mu_{t})d\mu_{t}+d[X]_{t}$$
$$=2(X_{t}-\mu_{t})dX_{t}-2(X_{t}-\mu_{t})d\mu_{t}+G^{2}dt.$$
So taking expectation leaves us with
$$Var(X_{t})=E[Y_{t}]=\int E [2(X_{t}-\mu_{t})F]dt-\int E[2(X_{t}-\mu_{t})]d\mu_{t}+\int E[G^{2}]dt,$$
where the Itô-integrals disappeared because they have mean zero Itō Integral has expectation zero.
So it remains t compute $\mu_{t}=E[X_{t}]$ and its t-derivative. To justify here swapping expectation and derivative, you can use When can we interchange the derivative with an expectation?. Since the F,G are unknown, this is as far as we can go.
