Importance sampling in path space for diffusion processes with slow-fast variables

TL;DR

This paper develops an importance sampling method for diffusion processes with slow and fast variables, providing bounds on estimator error and demonstrating effectiveness through numerical examples.

Contribution

It introduces a new importance sampling approach tailored for multiscale diffusion processes, leveraging limiting dynamics for control design.

Findings

Upper bound on relative error for importance sampling estimators

Control derived from limiting dynamics improves estimator efficiency

Numerical example validates the theoretical approach

Abstract

Importance sampling is a widely used technique to reduce the variance of a Monte Carlo estimator by an appropriate change of measure. In this work, we study importance sam- pling in the framework of diffusion process and consider the change of measure which is realized by adding a control force to the original dynamics. For certain exponential type expectation, the corresponding control force of the optimal change of measure leads to a zero-variance estimator and is related to the solution of a Hamilton-Jacobi-Bellmann equation. We focus on certain diffu- sions with both slow and fast variables, and the main result is that we obtain an upper bound of the relative error for the importance sampling estimators with control obtained from the limiting dynamics. We demonstrate our approximation strategy with an illustrative numerical example.