Importance Sampling for Multiscale Diffusions

TL;DR

This paper develops importance sampling schemes for multiscale stochastic differential equations with small noise, addressing inefficiencies of standard methods and providing conditions for asymptotic optimality with demonstrated examples.

Contribution

It introduces a novel importance sampling approach tailored for multiscale diffusions with small noise, ensuring asymptotic optimality and improved efficiency.

Findings

Schemes are asymptotically optimal under certain conditions

Examples demonstrate improved simulation efficiency

Addresses challenges of multiscale and small noise regimes

Abstract

We construct importance sampling schemes for stochastic differential equations with small noise and fast oscillating coefficients. Standard Monte Carlo methods perform poorly for these problems in the small noise limit. With multiscale processes there are additional complications, and indeed the straightforward adaptation of methods for standard small noise diffusions will not produce efficient schemes. Using the subsolution approach we construct schemes and identify conditions under which the schemes will be asymptotically optimal. Examples and simulation results are provided.