Importance Sampling for Pathwise Sensitivity of Stochastic Chaotic Systems

TL;DR

This paper introduces a new importance sampling-based pathwise sensitivity estimator for chaotic SDEs that maintains linear variance growth over time, improving computational efficiency and accuracy in sensitivity analysis.

Contribution

The paper develops a novel pathwise sensitivity estimator using a spring term and importance sampling, extending it with multilevel Monte Carlo and Richardson-Romberg extrapolation for enhanced efficiency.

Findings

Variance of the estimator grows linearly with time T

Estimator outperforms standard methods in chaotic SDEs

Numerical experiments confirm improved accuracy and efficiency

Abstract

This paper proposes a new pathwise sensitivity estimator for chaotic SDEs. By introducing a spring term between the original and perturbated SDEs, we derive a new estimator by importance sampling. The variance of the new estimator increases only linearly in time $T,$ compared with the exponential increase of the standard pathwise estimator. We compare our estimator with the Malliavin estimator and extend both of them to the Multilevel Monte Carlo method, which further improves the computational efficiency. Finally, we also consider using this estimator for the SDE with small volatility to approximate the sensitivities of the invariant measure of chaotic ODEs. Furthermore, Richardson-Romberg extrapolation on the volatility parameter gives a more accurate and efficient estimator. Numerical experiments support our analysis.