{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/adaptive-multilevel-monte-carlo-for","title":"Adaptive Multilevel Monte Carlo for Probabilities","arxiv_id":"2107.09148","date":"2021-07-19","proceeding":null,"authors":["Abdul-Lateef Haji-Ali","Jonathan Spence","Aretha Teckentrup"],"abstract":"We consider the numerical approximation of $\\mathbb{P}[G\\in \\Omega]$ where the $d$-dimensional random variable $G$ cannot be sampled directly, but there is a hierarchy of increasingly accurate approximations $\\{G_\\ell\\}_{\\ell\\in\\mathbb{N}}$ 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 improves this cost scaling slightly, but returns sub-optimal computational complexities since estimation of the probability involves a discontinuous functional of $G_\\ell$. We propose a general adaptive framework which is able to return the MLMC complexities seen for smooth or Lipschitz functionals of $G_\\ell$. 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 $G = \\mathbb{E}[X|Y]$ is approximated by an inner Monte Carlo estimate. Further experiments are given for digital option pricing, involving an approximation of a $d$-dimensional SDE.","url_abs":"https://arxiv.org/abs/2107.09148v1","url_pdf":"https://arxiv.org/pdf/2107.09148v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"abstracts"},"code_links":[{"paper_slug":"adaptive-multilevel-monte-carlo-for","repo_url":"https://github.com/JSpence97/mlmc-for-probabilities","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}