Abstract
In this case we have a finite state space E which we can take to be \(\left\{ {1,2,3,...,n} \right\}\). Given any q-matrix Q, which need not be conservative, there is a unique Q-function, and we can solve either the backward or forward equations to find it. Let us consider, for example, the backward equations
Suppose that Q is similar to the n x n matrix J; that is, that
for some n x n invertible matrix M. Then \(T\left( t \right) = {M^{ - 1}}P\left( t \right)\) satisfies
The point is J can in general be a simpler matrix than Q, with the result that the equation in (1.3) is straightforward to solve. The extreme case is when Q has n independent eigenvectors and thus J can be taken to be the diagonal matrix whose diagonal components are the eigenvalues of Q, and M is the matrix whose columns are the corresponding eigenvectors.
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© 1991 Springer-Verlag New York Inc.
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Anderson, W.J. (1991). Examples of Continuous-Time Markov Chains. In: Continuous-Time Markov Chains. Springer Series in Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-3038-0_3
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DOI: https://doi.org/10.1007/978-1-4612-3038-0_3
Publisher Name: Springer, New York, NY
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