Hostname: page-component-745bb68f8f-cphqk Total loading time: 0 Render date: 2025-02-11T07:19:20.008Z Has data issue: false hasContentIssue false
  • English
  • Français

Uniform approximation of the Cox–Ingersoll–Ross process via exact simulation at random times

Part of: Probabilistic methods, simulation and stochastic differential equations Stochastic analysis

Published online by Cambridge University Press:  11 January 2017

Grigori N. Milstein  and
John Schoenmakers
Grigori N. Milstein*
Affiliation:
Ural Federal University
John Schoenmakers*
Affiliation:
Weierstrass Institute for Applied Analysis and Stochastics
*
* Postal address: Ural Federal University, Lenin Str. 51, 620083 Ekaterinburg, Russia.
*– Postal address: Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39, 10117 Berlin, Germany. Email address: schoenma@wias-berlin.de
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we uniformly approximate the trajectories of the Cox–Ingersoll–Ross (CIR) process. At a sequence of random times the approximate trajectories will be even exact. In between, the approximation will be uniformly close to the exact trajectory. From a conceptual point of view, the proposed method gives a better quality of approximation in a path-wise sense than standard, or even exact, simulation of the CIR dynamics at some deterministic time grid.

Keywords

Cox–Ingersoll–Ross processSturm–Liouville problemBessel functionconfluent hypergeometric equation

MSC classification

Primary: 65C30: Stochastic differential and integral equations
Secondary: 60H35: Computational methods for stochastic equations
Type
Research Article
Information
Advances in Applied Probability , Volume 48 , Issue 4 , December 2016 , pp. 1095 - 1116
Copyright
Copyright © Applied Probability Trust 2017 

References

[1] Alfonsi, A. (2005). On the discretization schemes for the CIR (and Bessel squared) processes. Monte Carlo Meth. Appl. 11, 355–384.CrossRefGoogle Scholar
[2] Alfonsi, A. (2010). High order discretization schemes for the CIR process: application to affine term structure and Heston models. Math. Comput. 79, 209–237.CrossRefGoogle Scholar
[3] Andersen, L. (2008 ). Simple and efficient simulation of the Heston stochastic volatility model. J. Comput. Finance 11, 1–42.CrossRefGoogle Scholar
[4] Bateman, H.and Erdélyi, A. (1953). Higher Transcendental Functions. McGraw-Hill, New York.Google Scholar
[5] Beskos, A. and Roberts, G. O. (2005). Exact simulation of diffusions. Ann. Appl. Prob. 15, 2422–2444.CrossRefGoogle Scholar
[6] Beskos, A., Peluchetti, S. and Roberts, G. (2012). ε-strong simulation of the Brownian path. Bernoulli 18, 1223–1248.CrossRefGoogle Scholar
[7] Blanchet, J. and Murthy, K. R. A. (2014). Exact simulation of multidimensional reflected Brownian motion. Preprint. Available at http://arxiv.org/abs/1405.6469v2.pdf.Google Scholar
[8] Blanchet, J., Chen, X. and Dong, J. (2014). ε-strong simulation for multidimensional stochastic differential equations via rough path analysis. Preprint. Available at http://arxiv.org/abs/1403.5722v3.pdf.Google Scholar
[9] Broadie, M.and Kaya, Ö. (2006). Exact simulation of stochastic volatility and other affine jump diffusion processes. Operat. Res. 54, 217–231.CrossRefGoogle Scholar
[10] Chen, N. and Huang, Z. (2013). Localization and exact simulation of Brownian motion-driven stochastic differential equations. Math. Operat. Res. 38, 591–616.CrossRefGoogle Scholar
[11] Cox, J. C., Ingersoll, J. E., Jr. and Ross, S. A. (1985). A theory of the term structure of interest rates. Econometrica 53, 385–407.CrossRefGoogle Scholar
[12] Dereich, S., Neuenkirch, A. and Szpruch, L. (2012). An Euler-type method for the strong approximation of the Cox–Ingersoll–Ross process. Proc. R. Soc. London A 468, 1105–1115.Google Scholar
[13] Glasserman, P. (2004). Monte Carlo Methods in Financial Engineering Springer, New York.Google Scholar
[14] Göing-Jaeschke, A. and Yor, M. (2003). A survey and some generalizations of Bessel processes. Bernoulli 9, 313–349.CrossRefGoogle Scholar
[15] Heston, S. L. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev. Financial Studies 6, 327–343.CrossRefGoogle Scholar
[16] Higham, D. J. and Mao, X. (2005). Convergence of Monte Carlo simulations involving the mean-reverting square root process. J. Comput. Finance 8, 35–61.CrossRefGoogle Scholar
[17] Higham, D. J., Mao, X. and Stuart, A. M. (2002). Strong convergence of Euler-type methods for nonlinear stochastic differential equations. SIAM J. Numer. Anal. 40, 1041–1063.CrossRefGoogle Scholar
[18] Ikeda, N. and Watanabe, S. (1981). Stochastic Differential Equations and Diffusion Processes. North-Holland, Amsterdam.Google Scholar
[19] Itô, K. and McKean, H. P., Jr. (1974). Diffusion Processes and Their Sample Paths. Springer, Berlin.Google Scholar
[20] Karlin, S. and Taylor, H. M. (1981). A Second Course in Stochastic Processes. Academic Press, New York.Google Scholar
[21] Linetsky, V. (2004). Computing hitting time densities for CIR and OU diffusions: applications to mean-reverting models. J. Comput. Finance 7, 1–22.CrossRefGoogle Scholar
[22] Milstein, G. N. and Schoenmakers, J. (2015). Uniform approximation of the Cox–Ingersoll–Ross process. Adv. Appl. Prob. 47, 1132–1156.CrossRefGoogle Scholar
[23] Milstein, G. N. and Tretyakov, M. V. (1999). Simulation of a space-time bounded diffusion. Ann. Appl. Prob 9, 732–779.CrossRefGoogle Scholar
[24] Milstein, G. N. and Tretyakov, M. V. (2004). Stochastic Numerics for Mathematical Physics. Springer, Berlin.CrossRefGoogle Scholar
[25] Milstein, G. N. and Tretyakov, M. V. (2005). Numerical analysis of Monte Carlo evaluation of Greeks by finite differences. J. Comput. Finance 8, 1–33. CrossRefGoogle Scholar
[26] Revuz, D.and Yor, M. (1991). Continuous Martingales and Brownian Motion. Springer, Berlin.CrossRefGoogle Scholar
[27] Rogers, L. C. G. and Williams, D. (1987). Diffusions, Markov Processes, and Martingales, Vol.2, Itô Calculus. John Wiley, New York.Google Scholar