I am trying to derive the differential of the product of two processes, but I got stuck. This is what I have until now:
We have the following two stochastic processes: $dX_t= \mu_t dt +\sigma_t dW_t$ and $dY_t = \eta_t dt + \vartheta_t d \bar{W}_t$, with the correlation between the two Brownian motions equal to $\rho$, that is
E[($W_t - W_s$)($\bar{W}_t - \bar{W}_s$)|$F_s$] = $\rho(t-s)$ for s $\leq$ t.
Then I can get an expression for $d(X_t Y_t)$ with the following trick:
I start with decomposition: $(X_t+Y_t)^2=X_t^2+Y_t^2+ 2X_t Y_t$,
Which leads by differentiation to $d(X_t Y_t)= \frac{1}{2}[d(\{X_t +Y_t\}^2) - d(X_t^2) - d(Y_t^2)]$
Next I applied Ito-lemma to all three parts separately as follows: $d(X_t^2)=(2\mu_tXt+\sigma_t^2)dt+ 2\sigma_t X_tdW_t= 2X_t dX_t + \sigma_t^2dt$ $d(Y_t^2)=(2\eta_t Y_t+\vartheta_t^2)dt+ 2\vartheta_t Y_t d\bar{W}_t= 2Y_t dY_t+\vartheta_t^2 dt$
Now I don't know how to apply Ito lemma to the last part, i.e., $d(\{X_t +Y_t\}^2)$. Particularly, I don't know how to account for the correlation between the two Brownian Motions. Can someone help me with this last step?