 $$\begin{aligned} Q_0:={\mathbb E^{\widehat \P}\left[\mathcal N\left(d_0(t,S_t,X_{t,T}^\tau) \right)\bigg{|}S_t\right]} &= \int_0^{+\infty} \mathcal N\left( d_0(t,S_t,x) \right) f_{X_{t,T}^\tau}(x) {\mathrm d}x