  Under multiplicative drift and other regularity conditions, it is established
that the asymptotic variance associated with a particle filter approximation of
the prediction filter is bounded uniformly in time, and the nonasymptotic,
relative variance associated with a particle approximation of the normalizing
constant is bounded linearly in time. The conditions are demonstrated to hold
for some hidden Markov models on noncompact state spaces. The particle
stability results are obtained by proving $v$-norm multiplicative stability and
exponential moment results for the underlying Feynman-Kac formulas.
