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

by M. Hutzenthaler and A. Jentzen and X. Wang

(Report number 2013-34)

Abstract
Exponential integrability properties of numerical approximations are a key tool towards establishing positive rates of strong and numerically weak convergence for a large class of nonlinear stochastic differential equations; cf. Cox et al. [3]. 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.

Keywords: Exponential moments, numerical approximation, stochastic dierential equation, Euler scheme, Euler- Maruyama, implicit Euler scheme, tamed Euler scheme

@Techreport{HJW13_531,
  author = {M. Hutzenthaler and A. Jentzen and X. Wang},
  title = {Exponential integrability properties of numerical approximation processes for nonlinear stochastic differential equations},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2013-34},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2013/2013-34.pdf },
  year = {2013}
}

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