Importance sampling in path space for diffusion processes

Carsten Hartmann, Christof Schütte, Marcus Weber, Wei Zhang

Published 2015-02-27Version 1

Importance sampling is a widely used technique to reduce the variance of a Monte Carlo estimator by an appropriate change of measure. In this work, we study importance sampling in the framework of diffusion process and consider a change of measure which is realized by adding a control force to the original dynamics. For certain exponential type expectation, the corresponding control force of the optimal change of measure leads to a zero-variance estimator and is related to the solution of a Hamilton-Jacobi-Bellmann equation. We prove that for a certain class of multiscale diffusions, the control force obtained from the limiting dynamics is asymptotically optimal, and we provide an error bound for the importance sampling estimators under such suboptimal controls. We also discuss two other situations in which one can approximate the optimal control force by solving simplified dynamics. We demonstrate our approximation strategy with several numerical examples and discuss its application to large-scale systems, e.g. from molecular dynamics or material science.