How to extract transition matrices for smaller intervals than the original transition matrix?

Question

I have a transition matrix, for example:

\begin{bmatrix} 0.95 & 0.03 & 0.02 & 0 &0 \\ 0 &0.90 &0.1 &0 & 0\\ 0 &0.05 & 0.80 &0.1 &0.05 \\ 0& 0 & 0.05 & 0.90& 0.05\\ 0& 0 &0 &0.1 &0.9 \end{bmatrix}

Let us say this matrix represents the transition from a vector, for example: \begin{bmatrix} 22\\ 9\\ 13\\ 18\\ 10 \end{bmatrix} to another one, for an interval of 12 years.

I would like to know if it is possible to extract an annual transition matrix from this one. A repeated use of this annual transition matrix (12 times) should of course give me the same end vector.

I have had various ideas that failed including (don't laugh I am not a mathematics student):

• trying to find a matrix that if multiplying 11 times itself would equal the original matrix

• dividing the non-diagonal transitions by 12 and getting the row sums back to 1

• nth roots

To detect if it failed I simply applied any of my ideas and checked if the end vector was the same as obtained through the use of the original matrix.

Answer 1

For some necessary and some sufficient conditions, you might look at my paper with Jeff Rosenthal and Jason Wei, "Finding Generators for Markov Chains via Empirical Transition Matrices, with Applications to Credit Ratings", Mathematical Finance 11 (2001), 245-265.

I might note that no such transition matrix with nonnegative entries can exist in your example, but you can get close.

Answer 2

You can carry out an eigendecomposition of the transition matrix $T$ (using various statistical tools/packages, including R), so that you can express it as $$T=UDU^{-1}$$
where the $i^{th}$ column of square matrix $U$ is the $i^{th}$ eigenvector, and $D$ is a diagonal matrix whose entry in row $i$ and column $i$ is the corresponding eigenvalue $\lambda_i$.

Having done this, it is quite straightforward to calculate $T^n$, as this will be $$T^n=UD^{n}U^{-1}$$
For the annual transition matrix, you would set $n=\frac{1}{12}$, having found matrices $U$ (hence $U^{-1}$) and $D$.

However, as Robert Israel has usefully pointed out, certain conditions need to be satisfied for the entries of $T^n$ to be all positive, which does not hold for your transition matrix.