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Suppose I have a process X1, for which I do not have a generator matrix, only a transition (probability) matrix P1 for some time interval T, e.g. T=100. Suppose I have another process X2, such that X2 is identical to X1 except that if X2 transitions to a state below a certain threshold, it is returned to one of the states above the threshold (with equal probability of each) at 1 time period thereafter, where 1 is the finest granularity time interval under consideration.

My question is, would there be any way to estimate a new transition matrix P2 for X2 (over time period T) directly from the old one, without the need to worry about the so-called embedding problem for Markov chains (or solving a quadratic optimisation to infer the best generator)?

To illustrate, if there are 3 states a-c above the threshold and another 3 states d-f below, then my known 1-period transition probabilities would be:

      a   b    c   d    e    f
    a[ .    .    .    .  .  .,
    b  .    .    .    .  .  .,
    c  .    .    .    .  .  .,
    d  1/3  1/3  1/3  0  0  0,
    e  1/3  1/3  1/3  0  0  0,
    f  0     0    0   0  0  1];

where 'f' is an absorbing state. How would I use this information to obtain P2 for the T-period case, given that I know P1?

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Note: a unique generator may or may not exist for X1; I am only interested in obtaining a transition probability matrix P2 for X2. –  HungryStatistician Sep 3 '12 at 9:47
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