Rare event simulation for multiscale diffusions in random environments

TL;DR

This paper develops efficient importance sampling methods for estimating rare event probabilities in multiscale stochastic differential equations with random environments, addressing challenges posed by small noise, multiple scales, and environmental randomness.

Contribution

It introduces explicit measure changes that are proven to be asymptotically efficient in quenched settings for complex multiscale stochastic systems.

Findings

Importance sampling schemes are logarithmically asymptotically efficient.

Numerical simulations confirm theoretical efficiency results.

Method effectively handles randomness and multiple scales.

Abstract

We consider systems of stochastic differential equations with multiple scales and small noise and assume that the coefficients of the equations are ergodic and stationary random fields. Our goal is to construct provably-efficient importance sampling Monte Carlo methods that allow efficient computation of rare event probabilities or expectations of functionals that can be associated with rare events. Standard Monte Carlo algorithms perform poorly in the small noise limit and hence fast simulations algorithms become relevant. The presence of multiple scales complicates the design and the analysis of efficient importance sampling schemes. An additional complication is the randomness of the environment. We construct explicit changes of measures that are proven to be logarithmic asymptotically efficient with probability one with respect to the random environment (i.e., in the quenched sense). Numerical simulations support the theoretical results.