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

For a time-homogeneous Markov jump process with transition matrix P(t) and generator matrix A, which statement is correct?

Each row of the generator matrix sums to zero because the diagonal is minus the total exit rate, and the transition matrix satisfies both Kolmogorov equations, dP/dt = P(t)A and dP/dt = A P(t), giving P(t) = exp(tA).

  1. AEach row of A sums to zero, and P(t) satisfies dP/dt = P(t)A = A P(t)Correct
  2. BEach column of A sums to one, and P(t) = A t
  3. CEach row of A sums to one, and dP/dt = A + P(t)
  4. DEach row of A sums to zero, and P(t) = A^t
  5. The diagonal entries of A are positive and equal the sum of off-diagonal entries

Explanation

Diagonal entries are the negative of the total exit rate, so rows sum to zero. The forward and backward equations give dP/dt = P(t)A = A P(t), with solution P(t) = exp(tA). The other options misstate the row sums, the sign of the diagonal or the form of the solution.

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