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Numerical approximations of stochastic differential equations with non-globally Lipschitz continuous coefficients

About this Title

Martin Hutzenthaler, LMU Biozentrum, Department Biologie II, University of Munich (LMU), 82152 Planegg-Martinsried, Germany and Arnulf Jentzen, Seminar for Applied Mathematics, Swiss Federal Institute of Technology Zurich, 8092 Zurich, Switzerland; Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ 08544-1000, USA

Publication: Memoirs of the American Mathematical Society
Publication Year: 2015; Volume 236, Number 1112
ISBNs: 978-1-4704-0984-5 (print); 978-1-4704-2278-3 (online)
Published electronically: November 14, 2014
Keywords: Stochastic differential equation, rare event, strong convergence, numerical approximation, local Lipschitz condition, Lyapunov condition
MSC: Primary 60H35; Secondary 65C05, 65C30

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Table of Contents


  • 1. Introduction
  • 2. Integrability properties of approximation processes for SDEs
  • 3. Convergence properties of approximation processes for SDEs
  • 4. Examples of SDEs


Many stochastic differential equations (SDEs) in the literature have a superlinearly growing nonlinearity in their drift or diffusion coefficient. Unfortunately, moments of the computationally efficient Euler-Maruyama approximation method diverge for these SDEs in finite time. This article develops a general theory based on rare events for studying integrability properties such as moment bounds for discrete-time stochastic processes. Using this approach, we establish moment bounds for fully and partially drift-implicit Euler methods and for a class of new explicit approximation methods which require only a few more arithmetical operations than the Euler-Maruyama method. These moment bounds are then used to prove strong convergence of the proposed schemes. Finally, we illustrate our results for several SDEs from finance, physics, biology and chemistry.

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