Computing Hitting Probabilities of Markov Chains: Structural Results with regard to the Solution Space of the Corresponding System of Equations

Abstract

In a previous paper, we have shown that forward use of the steady-state difference equations arising from homogeneous discrete-state space Markov chains may be subject to inherent numerical instability. More precisely, we have proven that, under some appropriate assumptions on the transition probability matrix P, the solution space S of the difference equation may be partitioned into two subspaces $S=S_1\oplus S_2$, where the stationary measure of P is an element of $S_1$, and all solutions in $S_1$ are asymptotically dominated by the solutions corresponding to $S_2$. In this paper, we discuss the analogous problem of computing hitting probabilities of Markov chains, which is affected by the same numerical phenomenon. In addition, we have to fulfill a somewhat complicated side condition which essentially differs from those conditions one is usually confronted with when solving initial and boundary value problems. To extract the desired solution, an efficient and numerically stable generalized-continued-fraction-based algorithm is developed.