Finite transition matrices will be called A and also later M. They will have K rows corresponding to the K states of a Markov chain. Matrix entries are real, non-negative and each of the K rows of A sums to 1. For simplicity, it is assumed A has K distinct, possibly complex, eigenvalues: lamda1lamdaK.
For illustration only, in the examples and results below (i) K will equal 3 or 8 (ii) all the chains will have one absorbing state, that is A has one row with just one non-zero element equal to 1.
This post considers the general problem of given a transition matrix A find a transition matrix M such that M to the power n equals A. For illustration only , in the examples below, n will be 2 or 12 and the elements of A will be transition probabilities based on, possibly adjusted, annual published data. For those interested in the financial applications, to monthly (or quarterly) stub periods, John Hull¹s excellent book, ³Options ,Futures and Other Derivatives²,6th edition, 2006 provides more background on credit risk migration matrices .
In general, given a transition matrix A, there is not always an exact solution even of M.M = A where it is required that M be a transition matrix. Consider a transition matrix A with K = 3 and six non-zero entries at the (1,1),(1,2),(2,1),(2,2),(2,3) and (3,3) positions respectively 0.9, 0.1, 0.1, 0.8, 0.1 and 1.0. It is clear that no transition matrix M, with its associated chain, satisfies M.M = A: by considering the zero at the (1,3) position of A and thence the (1,2) and (2,3) entries in M. Does anyone know theoretical conditions for an exact solution to such equations, I assume unique, when one exists?
In general amazingly good approximate solutions, for example in the case of diagonally dominant K=8, credit migration matrices, involves using only the eigensystem of A and sensible rounding. I vaguely recall, the approach described below, was mentioned at an S&P presentation in London very many years ago, for credit migration matrices, A . Does anyone know appropriate book or paper references? Is this approach best in some norm sense? How does performance depend on (i) the diagonal dominance of A and (ii) sparseness of A?
If D is a matrix of normalised eigenvectors of A and InvD its inverse then InvD.A.D = Diag[lamda1,,lamdaK]: the diagonalised matrix of eigenvalues. Good approximate solutions to the general problem can be obtained from a matrix X where X = D. (Diag [lamda1,.., lamdaK] ^ (1/ n)).InvD. In general, X will not be a transition matrix or even real but replacing all negative and complex entries with 0 and adjusting the diagonal elements of N to produce row sums equal to 1 consistently produces an amazingly good M.
Finally I should perhaps note (i) the ³Homer Simpson² solution of dividing off-diagonal elements of A by n and adjusting (increasing) the diagonal elements so that row sums equal 1 (ii) the approaches to producing risk-neutral probabilities which may reduce diagonal dominance : see ³Synthetic CDOs², C.C.Mounfield, CUP,2009.