Adaptive Multilevel Monte Carlo for Probabilities

Abdul-Lateef Haji-Ali, Jonathan Spence, Aretha Teckentrup

Research output: Contribution to journalArticlepeer-review

3 Citations (Scopus)
41 Downloads (Pure)


We consider the numerical approximation of P[G ∊ Ω], where the d-dimensional random variable G cannot be sampled directly, but there is a hierarchy of increasingly accurate approximations {G } N which can be sampled. The cost of standard Monte Carlo estimation scales poorly with accuracy in this setup since it compounds the approximation and sampling cost. A direct application of multilevel Monte Carlo (MLMC) improves this cost scaling slightly, but returns suboptimal computational complexities since estimation of the probability involves a discontinuous functional of G . We propose a general adaptive framework which is able to return the MLMC complexities seen for smooth or Lipschitz functionals of G . Our assumptions and numerical analysis are kept general allowing the methods to be used for a wide class of problems. We present numerical experiments on nested simulation for risk estimation, where G = E [X| Y] is approximated by an inner Monte Carlo estimate. Further experiments are given for digital option pricing, involving an approximation of a d-dimensional SDE.

Original languageEnglish
Pages (from-to)2125-2149
Number of pages25
JournalSIAM Journal on Numerical Analysis
Issue number4
Early online date15 Aug 2022
Publication statusPublished - 2022


  • multilevel Monte Carlo
  • nested simulation
  • risk estimation

ASJC Scopus subject areas

  • Numerical Analysis
  • Computational Mathematics
  • Applied Mathematics


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