# Covariation of Ito semimartingales

If we have two Ito semimartingales over $$[0,T]$$: $$d X_t^i=a^i_tdt+\sigma_t^idW_t^i,\quad i=1,2$$ What is the relationship between $$\langle X^1,X^2 \rangle_t \quad \text{and} \quad \langle W^1,W^2 \rangle_t,$$ where $$\langle \rangle_t$$ denotes the quadratic variation? Assuming the correlation coefficient $$\rho$$ is constant, I think we should have $$\langle W^1,W^2 \rangle_t=\rho dt$$ while if it's not constant we would have $$\langle W^1,W^2 \rangle_t=\int_0^t\rho_s ds$$ How is this related to the the quadratic variation of $$X^1$$ and $$X^2$$, in the two cases? Is there any relationship with $$\int_0^t\rho_s \sigma_s^1 \sigma_s^2 ds$$?

Note that \begin{align*} \left\langle \int_0^t \sigma_s^1 dW_s^1, \int_0^t \sigma_s^2 dW_s^2\right\rangle &= \int_0^t \sigma_s^1 \sigma_s^2 d\langle W_s^1, W_s^2 \rangle\\ &=\int_0^t \rho_s\sigma_s^1 \sigma_s^2 ds. \end{align*}