Finding Generators for Markov Chains via Empirical Transition Matrices, with Applications to Credit Ratings

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Embeddable Markov Matrices (2010, background): “. Hence µr is the principal logarithm of λr and B is the principal logarithm of A. 7 The conclusions of the above corollary do not hold if A has repeated eigenvalues or a non-trivial Jordan form; see [3, 11]. For example the n×n identity matrix has a continuum of distinct logarithms B which do not a”
Analyzing rating transitions and rating drift with continuous observations (2002, background): “When estimating homogeneous chains in continuous-time by estimating the generator of the continuous-time Markov chain, we avoid the ‘embedding problem’ for Markov chains (for more on this in a rating modeling context, see Israel et al. (1999)).”
Measuring and Managing Credit Risk (2004, background): “Israel, Rosenthal, and Wie (2001) identify the conditions for the existence of a generator.”
Statistical inference for discretely observed Markov jump processes (2005, methodology): “This led Israel et al. (1997) to propose a method of estimating the jump intensities from discrete time observations.”