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

Which of the following processes is NOT, in general, a Markov chain without redefining its state space?

A process whose transition probabilities depend on both the current and the previous state is not a Markov chain on the original state space, because the history beyond the present matters. Redefining states as pairs could restore the property.

  1. AA process whose next-state probabilities depend on the current state and the previous stateCorrect
  2. BA simple random walk on the integers
  3. CA two-state healthy/sick model with fixed annual transition probabilities
  4. DA no-claims-discount process depending only on the current level
  5. A branching process where each generation depends only on the previous one

Explanation

If the next-state probabilities depend on two previous states, the Markov property fails for the original state variable. It can be made Markov only by redefining the state as the pair of last two states. The other processes depend only on the current state.

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