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IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Markov chains

Which statement about the Chapman-Kolmogorov equations for a time-homogeneous discrete-time Markov chain is correct?

The correct statement is that p_ij(m+n) equals the sum over intermediate states k of p_ik(m) times p_kj(n). It follows by conditioning on the state after m steps and using the Markov property, giving the matrix product P(m+n)=P(m)P(n).

  1. Ap_ij(m+n) = p_ij(m) + p_ij(n)
  2. Bp_ij(m+n) = sum over k of p_ik(m) p_kj(n)Correct
  3. Cp_ij(m+n) = p_ij(m) p_ij(n)
  4. Dp_ij(m+n) = sum over k of p_ik(m) p_ik(n)
  5. p_ij(m+n) = sum over k of p_ki(m) p_jk(n)

Explanation

Condition on the state k occupied after m steps; by the Markov property the remaining n steps depend only on k. This gives P(m+n) = P(m)P(n) in matrix form. The product of single entries (option 2) ignores the intermediate state sum.

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