\sigma_S \sqrt{X_{t,T}^\tau}} \bigg{|}\F_t^W \vee \F_{T-\tau}^P\right) \bigg{|}\widetilde \F_t\right]}\\ & =  {\mathbb E^{\widehat \P}\left[\mathcal N\left( d_0(t,S_t,X_{t,T}^\tau) \right) \bigg{|}\widetilde \F_t\right]}, \