IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Markov processes
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. Which expression is the Kolmogorov forward equation for P_HH(t)?
The forward equation is dP_HH/dt = -0.2 P_HH(t) + 0.6 P_HS(t). Probability leaves H at rate 0.2 and returns from S at rate 0.6, weighted by the probability of being in each state at time t.
- AdP_HH/dt = -0.2 P_HH(t) + 0.6 P_HS(t)Correct
- BdP_HH/dt = -0.6 P_HH(t) + 0.2 P_HS(t)
- CdP_HH/dt = 0.2 P_HH(t) - 0.6 P_HS(t)
- DdP_HH/dt = -0.2 P_HH(t) - 0.6 P_HS(t)
- dP_HH/dt = -0.8 P_HH(t) + 0.8 P_HS(t)
Explanation
The forward equation is dP_ij/dt = sum over k of P_ik(t) mu_kj. For j=H, i=H: P_HH(t)(-0.2) + P_HS(t)(0.6). The second option swaps the rates, which would be incorrect.
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