IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Stochastic processes
In a time-homogeneous Markov jump process, which statement about the generator matrix A with entries σij (i≠j) and σii is correct?
Rows of a generator matrix sum to zero, with non-negative off-diagonal rates and each diagonal entry equal to minus the total rate of leaving that state, so diagonals are non-positive. Row sums of one describe probability matrices instead.
- AEach row sums to one and all entries are non-negative
- BEach column sums to zero and diagonal entries are positive
- CEach row sums to zero, off-diagonal entries are non-negative and diagonal entries are non-positiveCorrect
- DEach row sums to zero and all entries are non-positive
- Diagonal entries equal the mean holding time of the state
Explanation
The generator has off-diagonal transition rates ≥ 0 and σii = −Σj≠i σij, so rows sum to zero and diagonals are ≤ 0. Row sums of one describe a transition probability matrix, not a generator.
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