Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 33 / 3 / a

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. PutThe inclusion of events and independence giveIn particular ${\sigma }_1$ is finite almost surely. For real $t\ge 0$, its tail is at most ${\rho }^{\lfloor t\rfloor }$. The tail integral formula for moments therefore gives, for every $p>0$,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 $p$.