 Solution

ID: past-exam-of-the-probability-statistics-and-options-ii-course-of-the-university-of-oxford/2011/m/m5/b/i/solution

Since refilling is immediate, the observed arrival-state space becomes

{1, \dots, N}

. A zero remainder is replaced by

N

, while all positive remainders are unchanged. Thus

P_{k N} = \frac{1}{k}, P_{kj} = \frac{1}{k} (1 \leq j < k), P_{kj} = 0 otherwise .

In particular state one moves to

N

with probability one, and state

N

1/ N

 New to topics? Read the docs here!