Let's suppose we have a random walk $$Z_t=\sum_{i=0}^{t} X_i$$ where each $X_i$ can take values $\pm 1$ with symmetric probability, $P(X=1)=P(X=-1)=\frac{1}{2}$.
It's is known that the hitting time at level $a$, $\sigma_a=\inf\{ t \geq 0: Z_t = a \}$, has an infinite expectation.
I am looking for help to compute the expected value $\tau = \sigma_a \land T$, where $T$ is a finite known time (therefore $\tau$ is a stopping time).
It makes sense that $E[\tau]\le T$ but I don't know how to continue from here.