Consider a markov chain with finite space { 0,1,..n} with transition probability matrix whose entries are $P_{ij}$. Let

$f_{ij}^n$ = probability that starting from state $i $ it goes to state $j$ first time.

Question 1. What are the necessary and sufficient condition arbitrary $a_{ij}^{n}$ needs to satisfy to be valid $f_{ij}^{n}$ of some markov chain ?

Qustion 2. If existence of such a markov chain is shown, can we calculate $P_{ij}$ given valid $f_{ij}^n$ ? ( I mean, is there any algorithm to calculate?)