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

A time-homogeneous Markov jump process has states A, B and C, with transition rates A→B = 0.2, A→C = 0.1, B→A = 0.3, B→C = 0.2 per year, and C absorbing. The process is in state A now. What is the probability it is still in state A at time h, to first order in h, given that it may have left and returned? Which statement is correct for small h?

P_AA(h) = 1 − 0.3h + o(h), because the force of leaving A is the sum of the rates to B and C, 0.2 + 0.1 = 0.3, and returning within a short interval requires two jumps, which is of order o(h).

  1. AP_AA(h) = 1 − 0.3h + o(h)Correct
  2. BP_AA(h) = 1 − 0.2h + o(h)
  3. CP_AA(h) = 1 − 0.5h + o(h)
  4. DP_AA(h) = 1 − 0.1h + o(h)
  5. P_AA(h) = 0.7h + o(h)

Explanation

The total rate out of A is 0.2 + 0.1 = 0.3, so P_AA(h) = 1 − 0.3h + o(h). Returning after leaving needs at least two jumps, which is o(h). The rate 0.5 wrongly uses B's exit rate, and 0.2 or 0.1 ignores one exit.

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