Suppose I have a random Gaussian walk $S_n=S_0 + \sum_{i=1}^n X_i$, with $S_0=0$ and where all $X_i$'s are normally distributed with mean $\mu$ and variance $\sigma^2$. Define the first exit time from a strip $[-a,a]$ as $\tau_a = \inf_n\{|S_n|\geq a\}$. I was wondering how one can then compute the mean exit time from the strip, i.e. $\mathbb{E}[\tau_{a}]$.
A similar question was asked here, but no definitive answer was provided. The TO stated that $\mathbb{E}[\tau_{a}] = a^2$, but my intuition says that in some sense it should depend on $\mu$ and $\sigma^2$. Could someone shed light on this matter?