formula

$1&=&E^{(\mu)}[\tilde{X}_t]\quad (0\leq t<\infty)\\ &=&E^{(\mu)}[E^{(\mu)}[\tilde{X}_t|\mathcal{F}^W_{t\wedge T_b}]]\\ &=&E^{(\mu)}[1_{\{T_b>t\}}\tilde{X}_t]+E^{(\mu)}[1_{\{T_b\leq t\}}\tilde{X}_{T_b}]$