State-dependent Importance Sampling for Estimating Expectations of Functionals of Sums of Independent Random Variables

arXiv (Cornell University)

Abstract

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.

Authors 4

1. Eya Ben Amar

   King Abdullah University of Science and Technology

   Affiliation as printed

   Computer, Electrical and Mathematical Sciences & Engineering Division (CEMSE) , King Abdullah University of Science and Technology (KAUST) , Thuwal , Saudi Arabia

2. Nadhir Ben Rached Aachen Chair of Mathematics for Uncertainty Quantification Department of Mathematics

   RWTH Aachen University

   Affiliation as printed

   Chair of Mathematics for Uncertainty Quantification , Department of Mathematics , RWTH Aachen University , Aachen , Germany

3. Abdul-Lateef Haji-Ali

   Heriot-Watt University

   Affiliation as printed

   School of Mathematical and Computer Sciences , Heriot-Watt University , Edinburgh , UK

4. Raul F. Tempone Aachen

   King Abdullah University of Science and Technology · RWTH Aachen University

   Affiliation as printed

   Alexander von Humboldt Professor in Mathematics for Uncertainty Quantification , RWTH Aachen University , Aachen , Germany

   Computer, Electrical and Mathematical Sciences & Engineering Division (CEMSE) , King Abdullah University of Science and Technology (KAUST) , Thuwal , Saudi Arabia