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What algorithm is Mathematica 9 using for FirstPassageTimeDistribution[]? Is it attempting to use PseudoInverse[] on the Laplacian of the graph underlying the discrete Markov process?

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I suggest you write to support@wolfram.com and ask. They will probably give you all the details you need (e.g. the paper reference). –  Szabolcs May 2 '13 at 21:15
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