In this work we employ importance sampling (IS) techniques to track a small over-threshold probability of a running maximum associated with the solution of a stochastic differential equation (SDE) within the framework of ensemble Kalman filtering (EnKF). Between two observation times of the EnKF, we propose to use IS with respect to the initial condition of the SDE, IS with respect to the Wiener process via a stochastic optimal control formulation, and combined IS with respect to both initial condition and Wiener process. Both IS strategies require the approximation of the solution of Kolmogorov Backward equation (KBE) with boundary conditions. In multidimensional settings, we employ a Markovian projection dimension reduction technique to obtain an approximation of the solution of the KBE by just solving a one dimensional PDE. The proposed ideas are tested on two illustrative examples: Double Well SDE and Langevin dynamics, and showcase a significant variance reduction compared to the standard Monte Carlo method and another sampling-based IS technique, namely, multilevel cross entropy.

翻译：本文在集合卡尔曼滤波（EnKF）框架下，采用重要性采样（IS）技术追踪与随机微分方程（SDE）解相关的运行最大值的超阈值概率。在EnKF的两个观测时间之间，我们提出三种IS策略：基于SDE初始条件的IS、通过随机最优控制公式对维纳过程实施的IS，以及结合初始条件和维纳过程的联合IS。两种IS策略均需近似求解带边界条件的科尔莫戈罗夫向后方程（KBE）。针对多维场景，我们采用马尔可夫投影降维技术，仅通过求解一维偏微分方程即可获得KBE解的近似。所提方法在两个示例（双阱SDE与朗之万动力学）中进行了验证，结果表明相较于标准蒙特卡洛方法及另一基于抽样的IS技术——多级交叉熵，该方法能实现显著的方差缩减。