Non-asymptotic performance analysis of importance sampling schemes for small noise diffusions

TL;DR

This paper provides a detailed prelimit analysis of importance sampling performance for small noise diffusions, using PDE-based asymptotic expansions to improve understanding of estimator variance and scheme efficiency.

Contribution

It introduces a PDE-based asymptotic expansion approach to analyze importance sampling performance beyond asymptotic limits, revealing sources of poor performance.

Findings

Derived a PDE characterization of the second moment in importance sampling.

Obtained explicit solutions for correction terms in the asymptotic expansion.

Identified factors affecting the performance of importance sampling schemes.

Abstract

In this note we develop a prelimit analysis of performance measures for importance sampling schemes related to small noise diffusion processes. In importance sampling the performance of any change of measure is characterized by its second moment. For a given change of measure, we characterize the second moment of the corresponding estimator as the solution to a PDE, which we analyze via a full asymptotic expansion with respect to the size of the noise and obtain a precise statement on its accuracy. The main correction term to the decay rate of the second moment solves a transport equation that can be solved explicitly. The asymptotic expansion that we obtain identifies the source of possible poor performance of nevertheless asymptotically optimal importance sampling schemes and allows for more accurate comparison among competing importance sampling schemes.