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IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Stochastic processes

A Markov chain on states {1,2,3,4} has transition probabilities p11 = 0.5, p12 = 0.5, p21 = 0.3, p22 = 0.7, p33 = 0.2, p34 = 0.8, p43 = 1. All other probabilities are zero. Which classification is correct?

The chain has two separate closed communicating classes, {1,2} and {3,4}, so it is not irreducible. Neither class can be left once entered, so both are recurrent, and the chain splits into two independent sub-chains.

  1. AThe chain is irreducible and aperiodic
  2. BThe chain has two closed communicating classes, {1,2} and {3,4}, and is not irreducibleCorrect
  3. CState 1 is transient because p12 > 0
  4. DThe chain has one closed class {1,2} and a transient class {3,4}
  5. The chain is irreducible with period 2

Explanation

From 1 and 2 the chain only reaches 1 and 2, so {1,2} is closed. From 3 and 4 it only reaches 3 and 4, so {3,4} is closed. No state in one set reaches the other, so there are two closed classes and the chain is not irreducible. Both classes are therefore recurrent, not transient.

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