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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. 87 (2018), 1353-1413
MSC (2010): Primary 60H35, 65C30
Published electronically: March 31, 2017
MathSciNet review: 3766391
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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
MR Author ID: 809631

Arnulf Jentzen
Affiliation: Department of Mathematics, ETH Zürich, Rämistrasse 101, 8092 Zürich, Switzerland
MR Author ID: 824543

Xiaojie Wang
Affiliation: School of Mathematics and Statistics, Central South University, Changsha 410083, Hunan, People’s Republic of China
MR Author ID: 898911

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