I have a finite space of s states {1, 2, …, s} and s fitted competing risks models given the process is in state i at time zero. This gives me estimates of probabilities of transitioning (first) to state j (j != i) by time t and not transitioning anywhere by time t (j = i). How should they be combined to determine the conditional (given i at time zero) probability of being in any state at time t?

If the time steps were discrete, I could raise the transition matrix to the appropriate power. Likewise, if the transition probabilities were simple functions of time, I could think about it in terms of an infinitesimal generator. However, I don’t see a “clean” way to do this in a general case.

Does anyone have a suggestion?

Thank you.