= Solution
The process $W_t=a^{-1}B_{a^2t}$ is standard <Brownian motion>, by <Brownian scaling>. Its first exit from $(-1,1)$ occurs at $\sigma_a/a^2$, so
$$
\sigma_a\ \stackrel{d}=\ a^2\sigma_1.
$$
The first part ensures integrability. Set $C=\mathbb E\sigma_1$; it is finite and strictly positive because path continuity at zero gives $\sigma_1>0$ almost surely. Therefore
$$
\boxed{\mathbb E\sigma_a=Ca^2\quad(a>0).}
$$
We keep $C$ unspecified until the comparison of limits requested below.
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