Importance sampling for slow-fast diffusions based on moderate deviations

TL;DR

This paper develops importance sampling schemes for slow-fast stochastic systems using moderate deviations, providing a more practical alternative to large deviations methods for estimating rare events in multiscale diffusions.

Contribution

It introduces asymptotically optimal importance sampling schemes based on moderate deviations for multiscale diffusions without periodicity assumptions, with explicit change of measure formulas.

Findings

Schemes are proven to be asymptotically optimal.

Explicit formulas for change of measure are derived.

Simulation results confirm theoretical effectiveness.

Abstract

We consider systems of slow--fast diffusions with small noise in the slow component. We construct provably logarithmic asymptotically optimal importance schemes for the estimation of rare events based on the moderate deviations principle. Using the subsolution approach we construct schemes and identify conditions under which the schemes will be asymptotically optimal. Moderate deviations--based importance sampling offers a viable alternative to large deviations importance sampling when the events are not too rare. In particular, in many cases of interest one can indeed construct the required change of measure in closed form, a task which is more complicated using the large deviations--based importance sampling, especially when it comes to multiscale dynamically evolving processes. The presence of multiple scales and the fact that we do not make any periodicity assumptions for the coefficients driving the processes, complicates the design and the analysis of efficient importance sampling schemes. Simulation studies illustrate the theory.