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I'm trying to understand a research paper but am facing a few difficulties, any help would be really appreciated.

A three state markov process with transition probabilities for states being:

0 --> -1 = λ(1-u)+s

0 --> 1 = λu

1 --> 0 = μ

I have a system of Kolmogorov Forward Equations

P'-1(t) = (λ(1-u)+s)P0(t)

P'0(t) = -(λ+s)P0(t)+ μP1(t)

P'1(t) = -μP1(t) + λuP0(t)

and I need to find a solution for P0(t) and P1(t) which is probability of the process being in either state 0/1 at time epoch 't'. I was wondering if anyone could go through how to obtain a solution for this system of equations. (If it is relevant the research paper utilises van Dantzigs' method of collective marks, however this is the one step which I do not follow)

Any relevant insight would be greatly appreciated!

Thanks.

Vivek
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    The question at http://stats.stackexchange.com/questions/46389/solving-the-kolmogorov-forward-equation-for-transition-probabilities seems to be the same as this (apart from the details of the process). Does it provide an adequate answer? – whuber Feb 28 '13 at 22:56
  • I have read through that and it is a very similar question however I have been unable to compute the solution to the KFE's with matricies, an example of how to solve the system detailed above would be excellent. – Vivek Mar 01 '13 at 13:46
  • You might want to research this question on the [math site](http://math.stackexchange.com/questions/10443), where it would be more on topic. Keywords to search for, if you want to do some research, are "eigenvector," "matrix diagonalization," and "differential equation." See what [Google](https://www.google.com/search?q=eigenvector+matrix+diagonalization+differential+equation) returns! – whuber Mar 01 '13 at 18:01

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