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Approximation of SDEs: a stochastic sewing approach
We give a new take on the error analysis of approximations of stochastic differential equations (SDEs), utilizing and developing the stochastic sewing lemma of Lê (Electron J Probab 25:55, 2020. 10.1214/20-EJP442). This approach allows one to exploit regularization by noise effects in obtaining conv...
Autores principales: | , , |
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Formato: | Online Artículo Texto |
Lenguaje: | English |
Publicado: |
Springer Berlin Heidelberg
2021
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Materias: | |
Acceso en línea: | https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8613171/ https://www.ncbi.nlm.nih.gov/pubmed/34898772 http://dx.doi.org/10.1007/s00440-021-01080-2 |
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author | Butkovsky, Oleg Dareiotis, Konstantinos Gerencsér, Máté |
author_facet | Butkovsky, Oleg Dareiotis, Konstantinos Gerencsér, Máté |
author_sort | Butkovsky, Oleg |
collection | PubMed |
description | We give a new take on the error analysis of approximations of stochastic differential equations (SDEs), utilizing and developing the stochastic sewing lemma of Lê (Electron J Probab 25:55, 2020. 10.1214/20-EJP442). This approach allows one to exploit regularization by noise effects in obtaining convergence rates. In our first application we show convergence (to our knowledge for the first time) of the Euler–Maruyama scheme for SDEs driven by fractional Brownian motions with non-regular drift. When the Hurst parameter is [Formula: see text] and the drift is [Formula: see text] , [Formula: see text] and [Formula: see text] , we show the strong [Formula: see text] and almost sure rates of convergence to be [Formula: see text] , for any [Formula: see text] . Our conditions on the regularity of the drift are optimal in the sense that they coincide with the conditions needed for the strong uniqueness of solutions from Catellier and Gubinelli (Stoch Process Appl 126(8):2323–2366, 2016. 10.1016/j.spa.2016.02.002). In a second application we consider the approximation of SDEs driven by multiplicative standard Brownian noise where we derive the almost optimal rate of convergence [Formula: see text] of the Euler–Maruyama scheme for [Formula: see text] drift, for any [Formula: see text] . |
format | Online Article Text |
id | pubmed-8613171 |
institution | National Center for Biotechnology Information |
language | English |
publishDate | 2021 |
publisher | Springer Berlin Heidelberg |
record_format | MEDLINE/PubMed |
spelling | pubmed-86131712021-12-10 Approximation of SDEs: a stochastic sewing approach Butkovsky, Oleg Dareiotis, Konstantinos Gerencsér, Máté Probab Theory Relat Fields Article We give a new take on the error analysis of approximations of stochastic differential equations (SDEs), utilizing and developing the stochastic sewing lemma of Lê (Electron J Probab 25:55, 2020. 10.1214/20-EJP442). This approach allows one to exploit regularization by noise effects in obtaining convergence rates. In our first application we show convergence (to our knowledge for the first time) of the Euler–Maruyama scheme for SDEs driven by fractional Brownian motions with non-regular drift. When the Hurst parameter is [Formula: see text] and the drift is [Formula: see text] , [Formula: see text] and [Formula: see text] , we show the strong [Formula: see text] and almost sure rates of convergence to be [Formula: see text] , for any [Formula: see text] . Our conditions on the regularity of the drift are optimal in the sense that they coincide with the conditions needed for the strong uniqueness of solutions from Catellier and Gubinelli (Stoch Process Appl 126(8):2323–2366, 2016. 10.1016/j.spa.2016.02.002). In a second application we consider the approximation of SDEs driven by multiplicative standard Brownian noise where we derive the almost optimal rate of convergence [Formula: see text] of the Euler–Maruyama scheme for [Formula: see text] drift, for any [Formula: see text] . Springer Berlin Heidelberg 2021-07-30 2021 /pmc/articles/PMC8613171/ /pubmed/34898772 http://dx.doi.org/10.1007/s00440-021-01080-2 Text en © The Author(s) 2021 https://creativecommons.org/licenses/by/4.0/Open AccessThis article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/ (https://creativecommons.org/licenses/by/4.0/) . |
spellingShingle | Article Butkovsky, Oleg Dareiotis, Konstantinos Gerencsér, Máté Approximation of SDEs: a stochastic sewing approach |
title | Approximation of SDEs: a stochastic sewing approach |
title_full | Approximation of SDEs: a stochastic sewing approach |
title_fullStr | Approximation of SDEs: a stochastic sewing approach |
title_full_unstemmed | Approximation of SDEs: a stochastic sewing approach |
title_short | Approximation of SDEs: a stochastic sewing approach |
title_sort | approximation of sdes: a stochastic sewing approach |
topic | Article |
url | https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8613171/ https://www.ncbi.nlm.nih.gov/pubmed/34898772 http://dx.doi.org/10.1007/s00440-021-01080-2 |
work_keys_str_mv | AT butkovskyoleg approximationofsdesastochasticsewingapproach AT dareiotiskonstantinos approximationofsdesastochasticsewingapproach AT gerencsermate approximationofsdesastochasticsewingapproach |