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Exponential integrability properties of numerical approximation processes for nonlinear stochastic differential equations


Authors: Martin Hutzenthaler, Arnulf Jentzen and Xiaojie Wang
Journal: Math. Comp.
MSC (2010): Primary 60H35, 65C30
DOI: https://doi.org/10.1090/mcom/3146
Published electronically: March 31, 2017
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Abstract: Exponential integrability properties of numerical approximations are a key tool for establishing positive rates of strong and numerically weak convergence for a large class of nonlinear stochastic differential equations. It turns out that well-known numerical approximation processes such as Euler-Maruyama approximations, linear-implicit Euler approximations, and some tamed Euler approximations from the literature rarely preserve exponential integrability properties of the exact solution. The main contribution of this article is to identify a class of stopped increment-tamed Euler approximations which preserve exponential integrability properties of the exact solution under minor additional assumptions on the involved functions.


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Additional Information

Martin Hutzenthaler
Affiliation: Fakultät für Mathematik, Universität Duisburg-Essen, Thea-Leymann-Strasse 9, 45127 Essen, Germany
Email: martin.hutzenthaler@uni-due.de

Arnulf Jentzen
Affiliation: Department of Mathematics, ETH Zürich, Rämistrasse 101, 8092 Zürich, Switzerland
Email: arnulf.jentzen@sam.math.ethz.ch

Xiaojie Wang
Affiliation: School of Mathematics and Statistics, Central South University, Changsha 410083, Hunan, People’s Republic of China
Email: x.j.wang7@csu.edu.cn

DOI: https://doi.org/10.1090/mcom/3146
Keywords: Exponential moments, numerical approximation, stochastic differential equation, Euler scheme, Euler-Maruyama, implicit Euler scheme, tamed Euler scheme, strong convergence rate
Received by editor(s): September 23, 2014
Received by editor(s) in revised form: June 9, 2015, October 8, 2015, February 3, 2016, and November 14, 2016
Published electronically: March 31, 2017
Additional Notes: The third author is the corresponding author
Article copyright: © Copyright 2017 American Mathematical Society