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Probability Theory and Related Fields

, Volume 170, Issue 1–2, pp 177–228 | Cite as

Importance sampling in path space for diffusion processes with slow-fast variables

  • Carsten Hartmann
  • Christof Schütte
  • Marcus Weber
  • Wei Zhang
Article

Abstract

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 the 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 focus on certain diffusions with both slow and fast variables, and the main result is that we obtain an upper bound of the relative error for the importance sampling estimators with control obtained from the limiting dynamics. We demonstrate our approximation strategy with an illustrative numerical example.

Keywords

Importance sampling Hamilton–Jacobi–Bellmann equation Monte Carlo method Change of measure Rare events Diffusion process 

Mathematics Subject Classification

Primary 60C05 60J60 Secondary 60F15 

Notes

Acknowledgements

The authors acknowledge financial support by the DFG Research Center Matheon, the Einstein Center for Mathematics ECMath and the DFG-CRC 1114 “Scaling Cascades in Complex Systems”. Special thanks also go to anonymous referees whose valuable comments and criticism have helped to improve this paper.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  • Carsten Hartmann
    • 1
  • Christof Schütte
    • 2
  • Marcus Weber
    • 2
  • Wei Zhang
    • 1
  1. 1.Institute of MathematicsFreie Universität BerlinBerlinGermany
  2. 2.Zuse Institute BerlinBerlinGermany

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