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

In this paper we identify conditions under which a true generator does or d oes not exist for an empirically observed Markov transition matrix. We show how to search for valid generators and choose the "correct" one that is th e most compatible with bond rating behaviors. We also show how to obtain an approximate generator when a true generator does not exist. We give illust rations using credit rating transition matrices published by Moody's and by Standard and Poor's.