Please refer to the exact same setting of this question:
Durret problem 4.8.3 - Random walk and optional sampling theorem application
I repeat the setting here in any case:
Let $S_n = \xi_1+\ldots+\xi_n$, $(\xi_i)_i$ independent with $0$ mean and $Var(\xi_i)=\sigma^2$. Then $S^2_n−n\sigma^2$ is a martingale. Let consider the stopping time $T=\inf \{n:|S_n|>a\}$.
Show $\mathbb{E}[T]>\frac{a^2}{\sigma^2}$.
Is there an expression that corresponds to $Var(T)$? Or at least an upper bound for $Var(T)$?
For example, if we are dealing with a symmetric simple random walk, then we can get an expression for $Var(T)$. See for example:
Stopping time computations via martingales
What about the more general case stated in the first link? I have tried to follow a similar approach, trying to come up with some clever martingale and apply the optional stopping theorem, but so far to no avail.
Thanks!