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?