IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Stochastic processes
A Markov jump process has states 1 (healthy), 2 (sick) and 3 (dead, absorbing). Constant rates: σ12 = 0.2, σ13 = 0.05, σ21 = 0.4, σ23 = 0.1 per year. Starting in state 1, which expression gives the probability that the first transition is to state 3 (death directly from healthy)?
The probability is 0.20. From state 1 the total exit rate is 0.25, and the chance the first jump goes to death is that transition rate, 0.05, divided by the total, giving 0.20.
- A0.05/0.25 = 0.20Correct
- B0.05/0.20 = 0.25
- C0.05/0.55 = 0.09
- D0.05 × 0.25 = 0.0125
- 0.05/0.10 = 0.50
Explanation
Total exit rate from state 1 is 0.2+0.05 = 0.25. The jump chain probability of going to 3 is 0.05/0.25 = 0.20. Dividing by 0.20 ignores the death rate in the total, and 0.55 wrongly includes rates out of state 2.
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