I am working on a problem where I now find myself wanting to apply Itô's formula to: \begin{equation} X_t = \exp(W_t -W_0-\frac{t}{2}+\int\limits_0^tX_sds) \end{equation} where $W_t$ is 1D Brownian motion and $X>0$. I usually start by setting up the partial derivatives of interest but here I am not sure how to think. Should I treat $X$, $t$ and $W$ as separate variables in Itô's formula?
Any advice would be much appreciated. Let me know if I am unclear.
When you are not sure, there is not harm in treating entries as separate variables to apply Ito's formula. The only thing you have to take care of in that case are quadratic covariations.