Embedding conditions for specific subsets of Markov transition matrices

The initial formulation of the continuous embedding problem by Elfving in 1937 concerns the question of whether for a discrete-time Markov chain there exists a compatible continuous-time Markov process, i.e., whether for a stochastic matrix P there exists an intensity matrix R that satisfies P=e^R. If P is the estimated transition matrix based on available data on unit time intervals, stochastic matrix roots A of P contribute to insights into the transition probabilities on shorter time intervals. The discrete embedding problem addresses the question of whether for a stochastic matrix P there exists a stochastic matrix A that satisfies P = A^n for n = 2,3, .... For both cases, continuous-time and discrete-time, necessary and sufficient embedding conditions have been investigated in previous work.

Depending on the context of the system under study, the transition matrix satisfies some specific properties and, therefore, belongs to a subset S of the set П of stochastic matrices. In that case, also the matrix roots are expected to reflect these specific properties. Therefore, the aim is to find necessary and sufficient conditions for a transition matrix P\in S to have a matrix root within S. We call this the discrete embedding problem for the subset S. Because in that case the discrete embedding problem, on the one hand, concerns a subset S of stochastic matrices P, and, on the other hand, imposes the additional condition that the matrix roots A must belong to S\subsetП, it is not obvious whether the embedding conditions for the subset S are less or more restrictive than the known embedding conditions for the set П of stochastic matrices. Therefore, further investigation is needed.

For example, Markov models for credit ratings are characterized by monotone credit rating transition matrices, where each row stochastically dominates each higher row. For the specific subset S of monotone transition matrices, the discrete embedding problem and embedding conditions are presented.