We have the martingale representation theorem: $G = E[G]+\int_0^t\theta_sdW_s$
Now, given $G=1_A$, where $A=\{exp(W_t)>K\}$, how to find the corresponding $\theta_t$?
The hint I received was to find $E[G]=P(A)=P(W_t>\ln K)$, then apply Ito's Lemma to the functional form of $E[G]$. And $E[G]$ should be a Gaussian integral. But I'm not sure how to apply Ito's Lemma to this integral.