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

• Published in: Journal of applied mathematics Vol. 2020; no. 2020; pp. 1 - 9
• Main Authors: Baumann, Hendrik, Hanschke, Thomas
• Format: Journal Article
• Language: English
• Published: Cairo, Egypt Hindawi Publishing Corporation 2020; Hindawi; John Wiley & Sons, Inc; Wiley
• Subjects: Algorithms; Boundary value problems; Chain stores; Computation; Difference equations; Markov chains; Markov processes; Solution space; Subspaces; Transition probabilities
• ISSN: 1110-757X; 1687-0042
• DOI: 10.1155/2020/9874072

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=S1⊕S2, where the stationary measure of P is an element of S1, and all solutions in S1 are asymptotically dominated by the solutions corresponding to S2. 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.