 Solution

ID: past-exam-of-the-probability-statistics-and-options-i-course-of-the-university-of-oxford/2010/m/2/b/solution

Condition on the full sampled history through time

n

X

implies that its future after

2 n

depends on that history only through

X_{2 n} = Y_{n}

. Summing over the intermediate state gives

P (Y_{n + 1} = j ∣ Y_{0}, \dots, Y_{n} = i) = \sum_{k} P_{ik} P_{kj} = (P^{2})_{ij} .

Thus $Y$ is a Markov chain with transition matrix $P^{2}$ . This is a deterministic-time sampled Markov chain.

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