Finding generators for Markov chains via empirical transition matrices, with applications to credit ratings (Q160231)

A stochastic matrix \(P\) is called embeddable if there exists a one-parameter semigroup \(t\to P(t)\), \(t\geq 0\), of stochastic matrices such that \(P(1)= P\). Any one-parameter semigroup of stochastic matrices may be written in the form \(P(t)= \exp(tQ)\), where \(Q\) is a generator, i.e. a matrix having row-sums \(0\) and nonnegative off-diagonal elements. The problem of embeddability and finding of a generator for a given empirical transition matrix is studied in the context of credit risk modelling. It is motivated by the following. The shortest time interval within which a transition matrix is estimated is typically one year. However, for valuation purposes it is required to know a transition matrix for a period shorter than one year. Many of the results concerning the embeddability problem are known in the theory of finite Markov chains.