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IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Transition intensities dependent on age (exact or census)

A healthy-sick-dead model has transition intensities independent of duration. The Kolmogorov forward equation for the probability of being healthy at age x+t, starting healthy at x, uses mu01 (healthy to sick), mu02 (healthy to dead) and mu10 (sick to healthy). Which expression for d/dt of tp00(x) is correct?

The correct equation is the derivative equals minus the total exit intensity from healthy times tp00, plus the sick-to-healthy intensity times tp01. Probability flows out of healthy through sickness and death, and flows back in from the sick state.

  1. A-(mu01 + mu02) × tp00 + mu10 × tp01Correct
  2. B-(mu01 + mu02) × tp00 - mu10 × tp01
  3. C(mu01 + mu02) × tp00 - mu10 × tp01
  4. D-(mu01 + mu02 + mu10) × tp00
  5. -mu01 × tp00 + mu10 × tp01

Explanation

The forward equation takes inflow into the state at time x+t minus outflow. Outflow from healthy is (mu01+mu02) × tp00. Inflow into healthy comes from sick at rate mu10 × tp01. The result is option A; the others have wrong signs, add mu10 to the outflow, or drop the death term.

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