IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Markov processes
A Markov jump process has states 1, 2, 3 with rates 1 to 2 of 0.3, 1 to 3 of 0.1, and state 3 absorbing. Rate 2 to 1 is 0.2 and 2 to 3 is 0.4. Starting in state 1, what is the expected time until the process first leaves state 1, and the probability that the first jump goes to state 3?
Expected holding time is 2.5 years, being 1 divided by the total exit rate 0.4, and the probability the first jump is to state 3 is 0.1 divided by 0.4, which equals 0.25.
- A2.5 years and 0.25Correct
- B2.5 years and 0.75
- C4.0 years and 0.25
- D0.4 years and 0.25
- 2.5 years and 0.10
Explanation
Total exit rate from 1 is 0.3+0.1=0.4, so mean holding time is 1/0.4=2.5 years. The first jump goes to 3 with probability 0.1/0.4=0.25. Option 0.75 is the probability of going to state 2 instead.
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