Sample Question #222 (mathematics – stochastics)

How would you prove that the probability of a random walk hitting any arbitrary number is 1?

(Hint: when you’re asked "how would you prove…", you’re expected to give a sketch of a proof, not the proof itself. Some interviewers would actually be pissed if you gave a formal proof, believing you didn’t understand what they were looking for. So, always answer the question in the form the interviewer wants — and this means you must listen to the questioning very carefully.)

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If there is a number (call it ‘a’) which is hit with probability <1 then it ‘a’ is NOT hit with positive probability. Since the paths of Brownian motion are continuous, with positive probability they stay bounded by ‘a’. But this is impossible because the variance must grow with time (sigma^2*t) which won’t happen if a path is bounded by ‘a’.