Let $B_t$ be a standard BM and define a stopping time $T := \{ \inf t \ge 0: B_t = -b+at^p \} $ for some $a,b>0$, $p>\frac{1}{2}$. Now it is claimed that $E(T)<\infty$. For the special case of $p=1$ a solution was given in Brownian Motion Hitting Time, however I don't see how to extend the argument there.
And given $E(T)<\infty$ is proven, can one deduce that for functions $f: [0,\infty) \to \mathbb{R}$ which satisfy $f(0)<0$ and $\frac{f(t)}{\sqrt{t}} \to \infty$ the stopping time $S := \{ \inf t \ge 0: B_t = f(t)\}$ has finite expectation?