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

A continuous-time Markov jump process has states 1 and 2 with transition rate 1→2 equal to 2 per year and 2→1 equal to 6 per year. A machine is initially in state 1. Which gives the limiting probability of being in state 1 and the approximate P(state 1 at time t) as t becomes large?

The limiting probability of state 1 is 0.75 and P11(t)=0.75+0.25e^{-8t}. Balance gives 2π1=6π2, so π1=0.75, and the transient term decays at the sum of the two rates, 2+6=8.

  1. ALimit 0.25, and P11(t)=0.25+0.75e^{-8t}
  2. BLimit 0.75, and P11(t)=0.75+0.25e^{-8t}Correct
  3. CLimit 0.75, and P11(t)=0.75+0.25e^{-4t}
  4. DLimit 0.25, and P11(t)=0.25+0.75e^{-4t}
  5. Limit 0.75, and P11(t)=0.75+0.25e^{8t}

Explanation

Balance: 2π1 = 6π2, so π1=0.75. The two-state process decays at rate λ+μ = 8, so P11(t)=π1+π2 e^{-8t}=0.75+0.25e^{-8t}. Using rate 4 is the mean of the rates, which is wrong, and a positive exponent diverges.

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