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Finance and Stochastics

, Volume 23, Issue 1, pp 139–172 | Cite as

On arbitrarily slow convergence rates for strong numerical approximations of Cox–Ingersoll–Ross processes and squared Bessel processes

  • Mario HefterEmail author
  • Arnulf Jentzen
Article
  • 111 Downloads

Abstract

Cox–Ingersoll–Ross (CIR) processes are extensively used in state-of-the-art models for the pricing of financial derivatives. The prices of financial derivatives are very often approximately computed by means of explicit or implicit Euler- or Milstein-type discretization methods based on equidistant evaluations of the driving noise processes. In this article, we study the strong convergence speeds of all such discretization methods. More specifically, the main result of this article reveals that each such discretization method achieves at most a strong convergence order of \(\delta /2\), where \(0<\delta <2\) is the dimension of the squared Bessel process associated to the considered CIR process.

Keywords

Cox–Ingersoll–Ross process Squared Bessel process Stochastic differential equation Strong (pathwise) approximation Lower error bound Optimal approximation 

Mathematics Subject Classification (2010)

60H10 65C30 

JEL Classification

C22 C63 G17 

Notes

Acknowledgements

Special thanks are due to André Herzwurm for a series of fruitful discussions on this work. This project has been supported through the SNSF-Research project 200021_156603 “Numerical approximations of nonlinear stochastic ordinary and partial differential equations”. We gratefully acknowledge the Institute for Mathematical Research (FIM) at ETH Zurich which provided office space and partially organized the short visit of the first author to ETH Zurich in 2016 when part of this work was done.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Fachbereich MathematikUniversität KaiserslauternKaiserslauternGermany
  2. 2.Seminar for Applied MathematicsEidgenössische Technische Hochschule ZürichZurichSwitzerland

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