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An adaptive splitting method for the Cox-Ingersoll-Ross process

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  • C'onall Kelly
  • Gabriel J. Lord

Abstract

We propose a new splitting method for strong numerical solution of the Cox-Ingersoll-Ross model. For this method, applied over both deterministic and adaptive random meshes, we prove a uniform moment bound and strong error results of order $1/4$ in $L_1$ and $L_2$ for the parameter regime $\kappa\theta>\sigma^2$. We then extend the new method to cover all parameter values by introducing a \emph{soft zero} region (where the deterministic flow determines the approximation) giving a hybrid type method to deal with the reflecting boundary. From numerical simulations we observe a rate of order $1$ when $\kappa\theta>\sigma^2$ rather than $1/4$. Asymptotically, for large noise, we observe that the rates of convergence decrease similarly to those of other schemes but that the proposed method making use of adaptive timestepping displays smaller error constants.

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  • C'onall Kelly & Gabriel J. Lord, 2021. "An adaptive splitting method for the Cox-Ingersoll-Ross process," Papers 2112.09465, arXiv.org, revised Feb 2023.
    • Handle: RePEc:arx:papers:2112.09465
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    1. Mario Hefter & Arnulf Jentzen, 2019. "On arbitrarily slow convergence rates for strong numerical approximations of Cox–Ingersoll–Ross processes and squared Bessel processes," Finance and Stochastics, Springer, vol. 23(1), pages 139-172, January.
    2. John C. Cox & Jonathan E. Ingersoll Jr. & Stephen A. Ross, 2005. "A Theory Of The Term Structure Of Interest Rates," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 5, pages 129-164, World Scientific Publishing Co. Pte. Ltd..
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    5. Alfonsi, Aurélien, 2013. "Strong order one convergence of a drift implicit Euler scheme: Application to the CIR process," Statistics & Probability Letters, Elsevier, vol. 83(2), pages 602-607.
    6. Jean-Francois Chassagneux & Antoine Jacquier & Ivo Mihaylov, 2014. "An explicit Euler scheme with strong rate of convergence for financial SDEs with non-Lipschitz coefficients," Papers 1405.3561, arXiv.org, revised Apr 2016.
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