Let $W_t$ be a standard Brownian Motion. I am curious under what conditions on a stopping time $\tau$ (of the filtration of this Brownian Motion) the expected value $\mathbb{E}[W_{\tau}]=\infty$? To be more precise, can somebody provide any examples of such stopping times? I am personally interested on how large such stopping times can be, i.e., are there any asymptotic bounds on $\mathbb{P}(\tau > t)$?
For example, we can take $\tau$ to be the first hitting time of a root boundary: $\tau = \inf(t: W_t \ge 1+\sqrt{t})$. This will follow from the fact that with probability one BM will hit such boundary, and since $\tau$ is at least as big as hitting level 1, the lower bound on the expected value will be infinity.
Are there any other examples of “lower” boundaries or maybe even much more complicated examples where stopping times are given not by a boundary? Any references will be appreciated!
The following paper by A. Novikov seems to provide an answer: Martingales, Tauberian Theorem, and Strategies of Gambling.