I have this exercise for uni where I need to rewrite a stochastic process, the exercise is as follows:
Let $\{(W_1(t), W_2(t)) : t ≥ 0\}$ be a two-dimensional Brownian motion defined on a probability space (Ω, F, P). Consider the process $\{Z(t) : t \geq 0\}$: $$Z(t) = 1 + e^{-W_1(t)}\int_0^t e^{W_1(u)}dW_2(u) $$ Prove that $Z(t)$ can be written as: $$Z(t) = 1+ W_2(t) - \int_0^t(Z(u) - 1)dW_1(u) + \frac{1}{2}\int_0^t(Z(u)-1)du$$
In this formula I recognise the Ito-Doeblin formula for an Ito process, but I don't quite see what the Ito process $X(t)$ and what the function $f(T,X(T))$ should be. I'm lacking some intuition with this subject matter as I've just recently started learning about it, so if anyone could push me in the right direction that would be greatly appreciated. Thanks in advance!!