Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 207 / 1 / f / i

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- Solution i

Solution

The transition probability matrix has entries

P_{rs} (u) = P (X (t + u) = s ∣ X (t) = r)

under the time-homogeneous Markov property. For the finite-state model in part d, the transition semigroup of a continuous-time Markov chain is

P (u) = e^{u Q} .

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