Large Deviations and Importance Sampling for Systems of Slow-Fast Motion

TL;DR

This paper develops a large deviations framework and optimal importance sampling schemes for multiscale stochastic systems with slow and fast dynamics, addressing efficiency issues in small noise regimes.

Contribution

It introduces a rigorous mathematical approach for importance sampling in dependent slow-fast stochastic systems, utilizing homogenization and control theory for asymptotic optimality.

Findings

Different behaviors depending on fast-slow interaction

Identification of asymptotically optimal importance sampling schemes

Standard Monte Carlo performs poorly in small noise, multiscale settings

Abstract

In this paper we develop the large deviations principle and a rigorous mathematical framework for asymptotically efficient importance sampling schemes for general, fully dependent systems of stochastic differential equations of slow and fast motion with small noise in the slow component. We assume periodicity with respect to the fast component. Depending on the interaction of the fast scale with the smallness of the noise, we get different behavior. We examine how one range of interaction differs from the other one both for the large deviations and for the importance sampling. We use the large deviations results to identify asymptotically optimal importance sampling schemes in each case. Standard Monte Carlo schemes perform poorly in the small noise limit. In the presence of multiscale aspects one faces additional difficulties and straightforward adaptation of importance sampling schemes for standard small noise diffusions will not produce efficient schemes. It turns out that one has to consider the so called cell problem from the homogenization theory for Hamilton-Jacobi-Bellman equations in order to guarantee asymptotic optimality. We use stochastic control arguments.