IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Markov processes
For a time-homogeneous Markov jump process with transition probabilities p_ij(t) and transition rates mu_ij, the Kolmogorov forward equation for d/dt p_01(t) is:
The forward equation is d/dt p_01(t) = sum over k≠1 of [p_0k(t) mu_k1 - p_01(t) mu_1k]. It balances the flow into state 1 from other states against the flow out of state 1 at time t.
- Asum over k≠1 of [p_0k(t) mu_k1 - p_01(t) mu_1k]Correct
- Bsum over k≠0 of [mu_0k p_k1(t) - mu_01 p_01(t)]
- Csum over k of p_0k(t) mu_1k
- Dsum over k≠1 of [p_0k(t) mu_1k - p_01(t) mu_k1]
- sum over k≠0 of [p_k1(t) mu_0k + p_01(t) mu_k0]
Explanation
The forward equation conditions on the most recent step: inflow into the destination state j=1 from other states k, p_0k(t) mu_k1, minus outflow from state 1, p_01(t) mu_1k. Option 1 is the backward equation, which conditions on the first step from the starting state.
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