= Solution
If $\sigma_1>n$, all the integer-time positions through $n$ lie in $(-1,1)$, so every increment over an intervening unit interval has absolute value at most two. The unit-time increments of <Brownian motion> are independent standard <normal random variables>. Put
$$
\rho=\mathbb P(|N(0,1)|\leq2)=2\Phi(2)-1\in(0,1).
$$
The inclusion of events and <independence> give
$$
\boxed{\mathbb P(\sigma_1>n)\leq\rho^n\quad(n\geq0).}
$$
In particular $\sigma_1$ is finite almost surely. For real $t\geq0$, its tail is at most $\rho^{\lfloor t\rfloor}$. The <tail integral formula for moments> therefore gives, for every $p>0$,
$$
\mathbb E\sigma_1^p=p\int_0^\infty t^{p-1}\mathbb P(\sigma_1>t)\,dt
\leq p\sum_{j=0}^\infty\rho^j\int_j^{j+1}t^{p-1}\,dt<\infty.
$$
An exponential geometric factor dominates the polynomial factors in this sum. This proves the requested moment finiteness for $p>1$, and in fact for every positive $p$.
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