In Karatzas and Shreve pg. 244 Chapter 4.2, it states the following equation:
$$u(a) = E^{a}f(W_{\tau_D}) = E^a\{E^a[f(W_{\tau_D})|\mathcal{F}_{\tau_{a+B_r}}]\} = E^a\{u(W_{\tau_{a+B_r}})\}$$
where he claims the last equality holds by Strong Markov Property. Note that $\tau_A := \inf\{t\geq 0 : W_t \in A^c\}$ and $u(x) := E^xf(W_{\tau_D})$, $x\in \bar{D}$. However, when I try to apply SMP, I arrive at:
$$E^a[f(W_{\tau_D})|\mathcal{F}_{\tau_{a+B_r}}](\omega) = E^{W_{\tau_{a+B_r}(\omega)}(\omega)}[f(W_{\tau_D(\omega) - \tau_{a+B_r}(\omega)}(\cdot))]$$
How can I get from here to $u(W_{\tau_{a+B_r}(\omega)}(\omega))$? I'm stuck because the expectation depends on $\omega$ inside but definition of $u(x)$ does not depend on $\omega$ inside.