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Topic: Finite Transition Matrix Equations: Some Easy Results & Sundry Questions
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Registered: 8/28/08
Finite Transition Matrix Equations: Some Easy Results & Sundry Questions
Posted: Jun 9, 2011 7:25 AM
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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: lamda1ŠlamdaK.

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,Š,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 CDOsŠ², C.C.Mounfield, CUP,2009.

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