Efficient Importance Sampling for Rare Event Simulation with Applications

TL;DR

This paper introduces a general method for optimal importance sampling in rare event simulation, providing explicit formulas for the tilting measure and demonstrating broad applicability across various distributions and practical financial and statistical problems.

Contribution

It develops a unified approach to find explicit optimal tilting measures for importance sampling, applicable to multiple distributions and real-world applications.

Findings

Explicit formulas for optimal tilting measures when the moment generating function exists

Applicable to normal, noncentral chi-squared, and compound Poisson distributions

Effective in computing value-at-risk and bootstrap confidence regions

Abstract

Importance sampling has been known as a powerful tool to reduce the variance of Monte Carlo estimator for rare event simulation. Based on the criterion of minimizing the variance of Monte Carlo estimator within a parametric family, we propose a general account for finding the optimal tilting measure. To this end, when the moment generating function of the underlying distribution exists, we obtain a simple and explicit expression of the optimal alternative distribution. The proposed algorithm is quite general to cover many interesting examples, such as normal distribution, noncentral $\chi^2$ distribution, and compound Poisson processes. To illustrate the broad applicability of our method, we study value-at-risk (VaR) computation in financial risk management and bootstrap confidence regions in statistical inferences.