 & = \widetilde S_t{\mathbb E^{\bar \P}\left[\mathcal N\left(d_0(t,S_t,X_{t,T}^\tau) \right)\bigg{|}\widetilde \F_t\right]}\nonumber\\ & = \widetilde S_t \frac{{\mathbb E^{\widehat \P}\left[\N\left(d_0(t,S_t,X_{t,T}^\tau) \right)e^{-\frac{\sigma_S^0}{0} \int_0^T P_{u-\tau}{\mathrm d}u + \sigma_S \int_0^T \sqrt{P_{u-\tau}}{\mathrm d}\widehat W_u}\bigg{|}\widetilde \F_t\right]}}{e^{-\frac{\sigma_S^0}{0} 