Let process $X_t$ have a stochastic differential, $dX_t = b_tdt + σ_tdW_t$.
Let $Y=(Y_t)_{t≥0}$ be the process defined as $Y_t=(1+t+W^2_t)X_t$. Find the SDE satisfied by the 2-dim process (Y, W).
I understand that you apply Ito's formula for a function $f(t,X_t)$. It is quite straightforward to calculate the first and second partial derivatives and apply the formula. However, that is strictly for function with $t$ and $X_t$ terms. In this case there is also a $W^2_t$ term present and I do not understand how to calculate the partial differential with respect to that. Would you treat this as an $X_t$ term in the funtion $f(t,x)$ or would you treat this as a constant?
The product rule to use is that $d(AB) = AdB + B dA + (dA)( dB).$ And then you use the rules $dt^2 = 0$, $dW dt = 0,$ and $dW^2 = dt$ to reduce.
So, for instance, for the term $W_t^2X_t,$ we get $$d(W_t^2X_t) = d(W_t^2) X_t + W_t^2 dX_t + d(W_t^2)dX_t. $$ And to break things down further, $$d(W_t^2) = 2 W_t dW_t + dW_t^2 = 2 W_t dW_t + dt .$$ So, for instance we have $$ d(W_t^2)dX_t = (2 W_t dW_t + dt)(b_t dt + \sigma_tdW_t) = 2W_t \sigma_t dt$$ for the third term of $d(W_t^2 X_t)$.
That's one piece of one term. Good luck with the rest.