  Estimating the expectations of functionals applied to sums of random
variables (RVs) is a well-known problem encountered in many challenging
applications. Generally, closed-form expressions of these quantities are out of
reach. A naive Monte Carlo simulation is an alternative approach. However, this
method requires numerous samples for rare event problems. Therefore, it is
paramount to use variance reduction techniques to develop fast and efficient
estimation methods. In this work, we use importance sampling (IS), known for
its efficiency in requiring fewer computations to achieve the same accuracy
requirements. We propose a state-dependent IS scheme based on a stochastic
optimal control formulation, where the control is dependent on state and time.
We aim to calculate rare event quantities that could be written as an
expectation of a functional of the sums of independent RVs. The proposed
algorithm is generic and can be applied without restrictions on the univariate
distributions of RVs or the functional applied to the sum. We apply this
approach to the log-normal distribution to compute the left tail and cumulative
distribution of the ratio of independent RVs. For each case, we numerically
demonstrate that the proposed state-dependent IS algorithm compares favorably
to most well-known estimators dealing with similar problems.
