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IAI Actuarial Core Principles · Actuarial Mathematics for Modelling · Valuing cashflows contingent on multiple transition events

In a time-homogeneous continuous-time Markov jump process with states Healthy (H), Sick (S) and Dead (D), the transition intensities are constant: H to S is 0.10, H to D is 0.02, S to H is 0.30, S to D is 0.06 (per year). What is the probability that a life currently Healthy will have left state H by a very small time interval h, to first order in h, by any route?

The probability is about 0.12h, because the total rate of leaving Healthy is the sum of the intensities to Sick (0.10) and Dead (0.02). Ignoring either route, or adding intensities from the Sick state, gives the wrong rate.

  1. A0.02h
  2. B0.10h
  3. C0.12hCorrect
  4. D0.30h
  5. 0.42h

Explanation

The total force of transition out of H is the sum of the intensities leaving H: 0.10 + 0.02 = 0.12. So P(leave H in h) is approximately 0.12h. Using only the H to S intensity (0.10h) ignores the death route. The 0.42h option wrongly adds the intensities leaving S.

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