  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 three illustrative examples: Double Well SDE,
Langevin dynamics and noisy Charney-deVore model, and showcase a significant
variance reduction compared to the standard Monte Carlo method and another
sampling-based IS technique, namely, multilevel cross entropy.
