  This article establishes sufficient conditions for a linear-in-time bound on
the non-asymptotic variance of particle approximations of time-homogeneous
Feynman-Kac formulae. These formulae appear in a wide variety of applications
including option pricing in finance and risk sensitive control in engineering.
In direct Monte Carlo approximation of these formulae, the non-asymptotic
variance typically increases at an exponential rate in the time parameter. It
is shown that a linear bound holds when a non-negative kernel, defined by the
logarithmic potential function and Markov kernel which specify the Feynman-Kac
model, satisfies a type of multiplicative drift condition and other regularity
assumptions. Examples illustrate that these conditions are general and flexible
enough to accommodate two rather extreme cases, which can occur in the context
of a non-compact state space: 1) when the potential function is bounded above,
not bounded below and the Markov kernel is not ergodic; and 2) when the
potential function is not bounded above, but the Markov kernel itself satisfies
a multiplicative drift condition.
