An iterative aggregation/disaggregation procedure for modelling the long-term behaviour of continuous-time evanescent random processes

Mark Bebbington and David Stewart

Abstract: We describe an aggregation/disaggregation method for finding the quasi-stationary distribution and decay parameters for continuous-time Markov chains. Finding the quasi-stationary distribution is equivalent to calculating the eigenvector corresponding to the smallest eigenvalue of the q-matrix restricted to the non-absorbing class. The method presented here is similar to an algebraic multigrid, with restriction operators that depend on the current approximation to the solution. The smoothers are short Arnoldi iterations or Gauss-Seidel iterations. Numerical results are presented for a variety of models of differing character, and indicate that the number of cycles requires grows only very slowly with the size of the problem.

Acknowledgement: This worked was funded by the Australian Research Council.

The authors:

• Mark Bebbington, Department of Statistics, School of Mathematical Information Sciences, Massey University. M.Bebbington@massey.ac.nz
• David Stewart, Department of Mathematics, Texas A&M University. Present address: Department of Mathematics, Virginia Tech. stewart@math.vt.edu

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Last modified: 5th February 1995