  We consider the computation of free energy-like quantities for diffusions in
high dimension, when resorting to Monte Carlo simulation is necessary. Such
stochastic computations typically suffer from high variance, in particular in a
low noise regime, because the expectation is dominated by rare trajectories for
which the observable reaches large values. Although importance sampling, or
tilting of trajectories, is now a standard technique for reducing the variance
of such estimators, quantitative criteria for proving that a given control
reduces variance are scarce, and often do not apply to practical situations.
The goal of this work is to provide a quantitative criterion for assessing
whether a given bias reduces variance, and at which order. We rely for this on
a recently introduced notion of stochastic solution for Hamilton-Jacobi-Bellman
(HJB) equations. Based on this tool, we introduce the notion of k-stochastic
viscosity approximation (SVA) of a HJB equation. We next prove that such
approximate solutions are associated with estimators having a relative variance
of order k-1 at log-scale. In particular, a sampling scheme built from a 1-SVA
has bounded variance as noise goes to zero. Finally, in order to show that our
definition is relevant, we provide examples of stochastic viscosity
approximations of order one and two, with a numerical illustration confirming
our theoretical findings.
