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

For a time-homogeneous Markov jump process with states 0, 1, 2 and transition rates, which of the following is the correct Kolmogorov forward equation for P00(t)? Assume rates are written as σij, with σ01 and σ02 the rates out of state 0, and σ10 and σ20 the rates into state 0 from states 1 and 2.

The forward equation is the derivative of P00 equal to minus P00 times the total exit rate from 0, plus P01 times σ10 plus P02 times σ20. It conditions on the state at time t and considers the last small time step.

  1. Ad/dt P00(t) = P00(t)(σ01+σ02) − P01(t)σ10 − P02(t)σ20
  2. Bd/dt P00(t) = −P00(t)(σ01+σ02) + P01(t)σ10 + P02(t)σ20Correct
  3. Cd/dt P00(t) = −P00(t)(σ10+σ20) + P01(t)σ01 + P02(t)σ02
  4. Dd/dt P00(t) = −P00(t)(σ01+σ02) + P10(t)σ01 + P20(t)σ02
  5. d/dt P00(t) = −P00(t)(σ01+σ02) + P01(t)σ10

Explanation

The forward equation conditions on the position at time t and looks at the final small interval: d/dt P0j(t) = Σk≠j Pk... precisely, P00 loses mass at rate σ01+σ02 and gains from states 1 and 2 at rates σ10 and σ20 weighted by P01 and P02. Option 3 uses the wrong rate set, and option 4 is the backward form.

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