Importance sampling for a robust and efficient multilevel Monte Carlo estimator for stochastic reaction networks

Chiheb Ben Hammouda*, Nadhir Ben Rached, Raúl Tempone

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

8 Scopus citations


The multilevel Monte Carlo (MLMC) method for continuous-time Markov chains, first introduced by Anderson and Higham (SIAM Multiscal Model Simul 10(1):146–179, 2012), is a highly efficient simulation technique that can be used to estimate various statistical quantities for stochastic reaction networks, in particular for stochastic biological systems. Unfortunately, the robustness and performance of the multilevel method can be affected by the high kurtosis, a phenomenon observed at the deep levels of MLMC, which leads to inaccurate estimates of the sample variance. In this work, we address cases where the high-kurtosis phenomenon is due to catastrophic coupling (characteristic of pure jump processes where coupled consecutive paths are identical in most of the simulations, while differences only appear in a tiny proportion) and introduce a pathwise-dependent importance sampling (IS) technique that improves the robustness and efficiency of the multilevel method. Our theoretical results, along with the conducted numerical experiments, demonstrate that our proposed method significantly reduces the kurtosis of the deep levels of MLMC, and also improves the strong convergence rate from β= 1 for the standard case (without IS), to β= 1 + δ, where 0 ' δ' 1 is a user-selected parameter in our IS algorithm. Due to the complexity theorem of MLMC, and given a pre-selected tolerance, TOL , this results in an improvement of the complexity from O(TOL - 2log (TOL) 2) in the standard case to O(TOL - 2) , which is the optimal complexity of the MLMC estimator. We achieve all these improvements with a negligible additional cost since our IS algorithm is only applied a few times across each simulated path.

Original languageEnglish (US)
Pages (from-to)1665-1689
Number of pages25
Issue number6
StatePublished - Nov 1 2020

Bibliographical note

Publisher Copyright:
© 2020, Springer Science+Business Media, LLC, part of Springer Nature.


  • Continuous-time Markov chains
  • Importance sampling
  • Multilevel Monte Carlo
  • Stochastic biological systems
  • Stochastic reaction networks

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Statistics and Probability
  • Statistics, Probability and Uncertainty
  • Computational Theory and Mathematics


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