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IAI Actuarial Core Principles · Actuarial Mathematics for Modelling · Valuing cashflows contingent on multiple transition events

For a time-homogeneous Markov jump process, the Kolmogorov forward equation for the probability of being in state j at time t, starting in i, expresses d/dt P_ij(t) as:

The forward equation says the rate of change of P_ij(t) equals inflow into j from other states, being P_ik(t) times the intensity from k to j, minus P_ij(t) times the total intensity out of j. The backward equation instead conditions on the first move from i.

  1. Athe sum over k of P_ik(t) times the intensity from k to j, minus P_ij(t) times the total intensity out of jCorrect
  2. Bthe sum over k of the intensity from i to k times P_kj(t) only
  3. CP_ij(t) multiplied by the total intensity into j
  4. Dthe sum over k of P_ik(t) times P_kj(t)
  5. the negative of the total intensity out of i multiplied by P_ij(t) only

Explanation

The forward equation balances inflow to j from other states k, weighted by the probability of being in k at time t, against outflow from j. The option using intensities from the starting state i with P_kj describes the backward equation. The product of probabilities is the Chapman-Kolmogorov equation.

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