IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Markov processes
A time-homogeneous Markov jump process has transition rates mu_ij (i not equal to j) and total exit rate from state i equal to -mu_ii. Which statement about the holding time in state i is correct?
The holding time in state i is exponentially distributed, with parameter equal to the total rate of leaving that state, the sum of the transition rates to all other states. This follows from the Markov memoryless property of the process.
- AIt is exponentially distributed with parameter equal to the total rate of leaving state iCorrect
- BIt is uniformly distributed on (0,1) for every state
- CIt is exponentially distributed with parameter equal to the number of states
- DIt follows a Poisson distribution with mean equal to the total rate of leaving state i
- It is normally distributed with mean equal to the total rate of leaving state i
Explanation
In a time-homogeneous Markov jump process the waiting time in state i is memoryless, so it is exponential. Its parameter is the sum of transition rates out of i. A Poisson or normal holding time would not be memoryless, and the number of states is irrelevant to the parameter.
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