I have question regarding this step:
Assume you have a stochastic process $X_t$ and a stopping time $\tau$.
Furthermore assume that some estimate like
$\mathbb{E}f(X_t)\leq \mathbb{E}g(X_t)$ holds for some suitable functions $f,g$.
Can I simply conclude that also
$\mathbb{E}f(X_t^\tau)\leq \mathbb{E}g(X_t^\tau)$
holds, where $X^\tau$ is the stopped process.