Butkovsky, O, Dareiotis, K and Gerencsér, M (2021) Approximation of SDEs: a stochastic sewing approach. Probability Theory and Related Fields, 181 (4). pp. 975-1034. ISSN 0178-8051
Abstract
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. https://doi.org/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 H∈(0,1) and the drift is Cα, α∈[0,1] and α>1−1/(2H), we show the strong Lp and almost sure rates of convergence to be ((1/2+αH)∧1)−ε, for any ε>0. 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. https://doi.org/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 1/2−ε of the Euler–Maruyama scheme for Cα drift, for any ε,α>0.
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Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © The Author(s) 2021. This 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/. |
Keywords: | Fractional Brownian motion; Irregular drift; Regularization by noise; Stochastic differential equations; Strong rate of convergence |
Dates: |
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Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Mathematics (Leeds) > Statistics (Leeds) |
Depositing User: | Symplectic Publications |
Date Deposited: | 20 Aug 2021 09:28 |
Last Modified: | 25 Jun 2023 22:44 |
Status: | Published |
Publisher: | Springer |
Identification Number: | 10.1007/s00440-021-01080-2 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:177279 |
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