  We give a development of the ODE method for the analysis of recursive
algorithms described by a stochastic recursion. With variability modelled via
an underlying Markov process, and under general assumptions, the following
results are obtained: 1. Stability of an associated ODE implies that the
stochastic recursion is stable in a strong sense when a gain parameter is
small. 2. The range of gain-values is quantified through a spectral analysis of
an associated linear operator, providing a non-local theory. 3. A second-order
analysis shows precisely how variability leads to sensitivity of the algorithm
with respect to the gain parameter.
  All results are obtained within the natural operator-theoretic framework of
geometrically ergodic Markov processes.
