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.

翻译：我们考虑在高维扩散中计算类自由能量时，必须借助蒙特卡罗模拟的情形。此类随机计算通常存在高方差问题，尤其在低噪声条件下，因为期望值由可达大值的稀有轨迹主导。尽管重要性采样（或轨迹倾斜）已成为降低此类估计量方差的常规技术，但能够证明特定控制可减少方差的定量判据却十分稀缺，且往往不适用于实际场景。本研究旨在提供一种定量判据，用以评估特定偏置能否以何种阶数降低方差。我们基于近期引入的Hamilton-Jacobi-Bellman（HJB）方程随机解概念展开工作：首先提出HJB方程的k阶随机黏性逼近（SVA）定义；进而证明此类近似解对应的估计量在对数尺度下具有k-1阶的相对方差。特别地，由1-SVA构建的采样方案在噪声趋零时方差有界。最后，为验证定义的实用性，我们给出了一阶与二阶随机黏性逼近的实例，并通过数值实验印证了理论发现。