IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Markov chains
A chain on states {1,2,3,4} has transition matrix rows: 1→(0.5,0.5,0,0); 2→(0.5,0.5,0,0); 3→(0.25,0,0.25,0.5); 4→(0,0,0.5,0.5). Which classification is correct?
{1,2} is a closed recurrent class, while {3,4} is transient. From state 3 there is a 0.25 chance of moving to state 1, from which return is impossible, and state 4 communicates with 3, so both leak into the closed class.
- A{1,2} is a closed recurrent class and {3,4} is transientCorrect
- BAll four states are recurrent
- C{1,2} is closed and recurrent, {3,4} is a closed recurrent class
- DStates 3 and 4 are absorbing
- {3,4} is transient and {1,2} is transient
Explanation
States 1 and 2 only move between themselves, so {1,2} is closed and recurrent. State 3 moves to state 1 with probability 0.25 and never returns from {1,2}, so 3 is transient. State 4 reaches 3, so it communicates with 3 and is also transient. Hence {3,4} is transient.
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