On arbitrarily slow convergence rates for strong numerical approximations of Cox–Ingersoll–Ross processes and squared Bessel processes
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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.
KeywordsCox–Ingersoll–Ross process Squared Bessel process Stochastic differential equation Strong (pathwise) approximation Lower error bound Optimal approximation
Mathematics Subject Classification (2010)60H10 65C30
JEL ClassificationC22 C63 G17
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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