IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Markov processes
In a time-homogeneous Markov jump process, the jump chain has transition probability from state i to j equal to:
The jump chain probability from i to j is mu_ij divided by the total rate of leaving i, the sum of mu_ik over k≠i. The next state is chosen in proportion to the transition rates out of the current state.
- Amu_ij multiplied by the holding time in i
- Bmu_ij divided by the sum over k≠i of mu_ikCorrect
- Cmu_ij divided by mu_jj
- Dp_ij(t) as t tends to zero
- mu_ij divided by the sum over k≠j of mu_kj
Explanation
On leaving state i, the next state is j with probability proportional to mu_ij, so the probability is mu_ij divided by the total exit rate from i, the sum of mu_ik over k≠i. Option 4 sums the inflow rates into j, which is the wrong direction.
Did you get it right without looking?
One question tells you little. A timed set on Markov processes shows your real accuracy, how long you take and where you lose marks.
More Markov processes questions
- A two-state Markov jump process has states H (healthy) and S (sick), with constant transition rates H to S of 0.2 and S to H of 0.6 per year…
- A discrete-time Markov chain on states {1,2,3,4} has transition matrix rows: state 1: (0.5, 0.5, 0, 0); state 2: (0.4, 0.6, 0, 0); state 3: …
- A time-homogeneous Markov jump process has states A, B and C, with transition rates A→B = 0.2, A→C = 0.1, B→A = 0.3, B→C = 0.2 per year, and…
- A two-state chain (states A and B) has P(A→B) = 0.2 and P(B→A) = 0.4. What is the long-run stationary probability of being in state A?
- 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…
- In a time-homogeneous Markov jump process with states Healthy (H), Sick (S) and Dead (D), the transition rates are constant. Which statement…