 \E^{\P^{\alpha, \beta}_F} \left[ \phi\left(B_{t_0\wedge \tau}, \dots, B_{t_n\wedge \tau}\right)\psi_{\tau}\right] = \E^{\P_{0,F}} \left[ \phi\Big( X_{t_0\wedge \widetilde{\tau}}^{\alpha,\beta}, \dots, X_{t_n\wedge \widetilde{\tau}}^{\alpha,\beta} \Big)