Use Ito lemma to solve some SDE

Question

Take B$_t$ to be a Brownian motion and Z$_t$ = e$^{\int_{0}^{t}g(s,w)dB_s-\frac{1}{2}\int_{0}^{t}g^2(s,w)ds}$, how can we apply Ito lemma to get SDE of Z$_t$?

I set Z$_t$ = f(t, B$_s$), then by Ito lemma, it follows that

dZ$_t$ = f$_t$dt + f$_{x}$dB$_s$ + $\frac{1}{2}$f$_{xx}$dt, but in our case, how to obtain f$_t$, f$_x$, f$_{xx}$ due to the existence of Ito integral in the formula?

Answer 1

You should write $Z_t=e^{Y_t}$ and notice that you know the SDE of $Y_t$ as it is provided in integral form. Then you can apply Ito's lemma on the exponential function.