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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 $\int_0^t\rho_s \sigma_s^1 \sigma_s^2 ds$?

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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*}

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