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


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.


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 



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.


  1. 1.
    Alfonsi, A.: On the discretization schemes for the CIR (and Bessel squared) processes. Monte Carlo Methods Appl. 11, 355–384 (2005) MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Alfonsi, A.: Strong order one convergence of a drift implicit Euler scheme: application to the CIR process. Stat. Probab. Lett. 83, 602–607 (2013) MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Berkaoui, A., Bossy, M., Diop, A.: Euler scheme for SDEs with non-Lipschitz diffusion coefficient: strong convergence. ESAIM Probab. Stat. 12, 1–11 (2008) MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Borodin, A.N., Salminen, P.: Handbook of Brownian Motion—Facts and Formulae, 2nd edn. Birkhäuser, Basel (2002) CrossRefzbMATHGoogle Scholar
  5. 5.
    Bossy, M., Olivero, H.: Strong convergence of the symmetrized Milstein scheme for some CEV-like SDEs. Bernoulli 24, 1995–2042 (2018) MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chassagneux, J.F., Jacquier, A., Mihaylov, I.: An explicit Euler scheme with strong rate of convergence for financial SDEs with non-Lipschitz coefficients. SIAM J. Financ. Math. 7, 993–1021 (2016) MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cox, J.C., Ingersoll, J.E. Jr., Ross, S.A.: A theory of the term structure of interest rates. Econometrica 53, 385–407 (1985) MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cozma, A., Reisinger, C.: Exponential integrability properties of Euler discretization schemes for the Cox–Ingersoll–Ross process. Discrete Contin. Dyn. Syst., Ser. B 21, 3359–3377 (2016) MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Cozma, A., Reisinger, C.: Strong order 1/2 convergence of full truncation Euler approximations to the Cox–Ingersoll–Ross process. Working paper (2017). Available online at arXiv:1704.07321
  10. 10.
    Deelstra, G., Delbaen, F.: Convergence of discretized stochastic (interest rate) processes with stochastic drift term. Appl. Stoch. Models Data Anal. 14, 77–84 (1998) MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Dereich, S., Neuenkirch, A., An, S.L.: Euler-type method for the strong approximation of the Cox–Ingersoll–Ross process. Proc. R. Soc. A, Math. Phys. Eng. Sci. 468, 1105–1115 (2012) MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Gerencsér, M., Jentzen, A., Salimova, D.: On stochastic differential equations with arbitrarily slow convergence rates for strong approximation in two space dimensions. Proc. R. Soc. A, Math. Phys. Eng. Sci. 473, 20170104 (2017) MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Göing-Jaeschke, A., Yor, M.: A survey and some generalizations of Bessel processes. Bernoulli 9, 313–349 (2003) MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Gyöngy, I.: A note on Euler’s approximations. Potential Anal. 8, 205–216 (1998) MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Gyöngy, I., Rásonyi, M.: A note on Euler approximations for SDEs with Hölder continuous diffusion coefficients. Stoch. Process. Appl. 121, 2189–2200 (2011) CrossRefzbMATHGoogle Scholar
  16. 16.
    Hairer, M., Hutzenthaler, M., Jentzen, A.: Loss of regularity for Kolmogorov equations. Ann. Probab. 43, 468–527 (2015) MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Hefter, M., Herzwurm, A.: Optimal strong approximation of the one-dimensional squared Bessel process. Commun. Math. Sci. 15, 2121–2141 (2017) MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Hefter, M., Herzwurm, A.: Strong convergence rates for Cox–Ingersoll–Ross processes—full parameter range. J. Math. Anal. Appl. 459, 1079–1101 (2018) MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Hefter, M., Herzwurm, A., Müller-Gronbach, T.: Lower error bounds for strong approximation of scalar SDEs with non-Lipschitzian coefficients. Ann. Appl. Probab. (2017). Available online at. Google Scholar
  20. 20.
    Higham, D., Mao, X.: Convergence of Monte Carlo simulations involving the mean-reverting square root process. J. Comput. Finance 8(3), 35–61 (2005) CrossRefGoogle Scholar
  21. 21.
    Higham, D.J., Mao, X., Stuart, A.M.: Strong convergence of Euler-type methods for nonlinear stochastic differential equations. SIAM J. Numer. Anal. 40, 1041–1063 (2002) MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Hofmann, N., Müller-Gronbach, T., Ritter, K.: The optimal discretization of stochastic differential equations. J. Complex. 17(1), 117–153 (2001) MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Hu, Y.: Semi-implicit Euler–Maruyama scheme for stiff stochastic equations. In: Körezlioğlu, H., et al. (eds.) Stochastic Analysis and Related Topics, V. Progr. Probab., vol. 38, pp. 183–202. Birkhäuser Boston, Boston (1996) CrossRefGoogle Scholar
  24. 24.
    Hutzenthaler, M., Jentzen, A.: Numerical approximations of stochastic differential equations with non-globally Lipschitz continuous coefficients. Mem. Am. Math. Soc. 236(1112), 1–99 (2015) MathSciNetzbMATHGoogle Scholar
  25. 25.
    Hutzenthaler, M., Jentzen, A., Kloeden, P.E.: Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients. Ann. Appl. Probab. 22, 1611–1641 (2012) MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Hutzenthaler, M., Jentzen, A., Noll, M.: Strong convergence rates and temporal regularity for Cox–Ingersoll–Ross processes and Bessel processes with accessible boundaries. Working paper (2014). Available online at arXiv:1403.6385
  27. 27.
    Jentzen, A., Müller-Gronbach, T., Yaroslavtseva, L.: On stochastic differential equations with arbitrary slow convergence rates for strong approximation. Commun. Math. Sci. 14, 1477–1500 (2016) MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Kallenberg, O.: Foundations of Modern Probability, 2nd edn. Springer, New York (2002) CrossRefzbMATHGoogle Scholar
  29. 29.
    Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus, 2nd edn. Springer, New York (1991) zbMATHGoogle Scholar
  30. 30.
    Kelly, C., Lord, G.J.: Adaptive time-stepping strategies for nonlinear stochastic systems. IMA J. Numer. Anal. 38(3), 1523–1549 (2018) MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Mao, X.: Stochastic Differential Equations and Their Applications. Horwood, Chichester (1997) zbMATHGoogle Scholar
  32. 32.
    Milstein, G.N., Schoenmakers, J.: Uniform approximation of the Cox–Ingersoll–Ross process. Adv. Appl. Probab. 47, 1132–1156 (2015) MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Müller-Gronbach, T.: Optimal pointwise approximation of SDEs based on Brownian motion at discrete points. Ann. Appl. Probab. 14, 1605–1642 (2004) MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Neuenkirch, A., Szpruch, L.: First order strong approximations of scalar SDEs defined in a domain. Numer. Math. 128, 103–136 (2014) MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion, 3rd edn. Springer, Berlin (1999) CrossRefzbMATHGoogle Scholar
  36. 36.
    Sabanis, S.: A note on tamed Euler approximations. Electron. Commun. Probab. 18, 47 (2013) MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Sabanis, S.: Euler approximations with varying coefficients: the case of superlinearly growing diffusion coefficients. Ann. Appl. Probab. 26, 2083–2105 (2016) MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Yaroslavtseva, L.: On non-polynomial lower error bounds for adaptive strong approximation of SDEs. J. Complex. 42, 1–18 (2017) MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Yaroslavtseva, L., Müller-Gronbach, T.: On sub-polynomial lower error bounds for quadrature of SDEs with bounded smooth coefficients. Stoch. Anal. Appl. 35, 423–451 (2017) MathSciNetCrossRefzbMATHGoogle Scholar

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

Personalised recommendations