Advances in Applied Probability

Uniform approximation of the Cox-Ingersoll-Ross process

Grigori N. Milstein and John Schoenmakers

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The Doss-Sussmann (DS) approach is used for uniform simulation of the Cox-Ingersoll-Ross (CIR) process. The DS formalism allows us to express trajectories of the CIR process through solutions of some ordinary differential equation (ODE) depending on realizations of a Wiener process involved. By simulating the first-passage times of the increments of the Wiener process to the boundary of an interval and solving the ODE, we uniformly approximate the trajectories of the CIR process. In this respect special attention is payed to simulation of trajectories near 0. From a conceptual point of view the proposed method gives a better quality of approximation (from a pathwise point of view) than standard, even exact, simulation of the stochastic differential equation at some deterministic time grid.

Article information

Adv. in Appl. Probab., Volume 47, Number 4 (2015), 1132-1156.

First available in Project Euclid: 11 December 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 65C30: Stochastic differential and integral equations
Secondary: 60H35: Computational methods for stochastic equations [See also 65C30]

Cox-Ingersoll-Ross process Doss-Sussmann formalism Bessel function confluent hypergeometric equation


Milstein, Grigori N.; Schoenmakers, John. Uniform approximation of the Cox-Ingersoll-Ross process. Adv. in Appl. Probab. 47 (2015), no. 4, 1132--1156. doi:10.1239/aap/1449859803.

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  • Alfonsi, A. (2005). On the discretization schemes for the CIR (and Bessel squared) processes. Monte Carlo Methods Appl. 11, 355–384.
  • Alfonsi, A. (2010). High order discretization schemes for the CIR process: application to affine term structure and Heston models. Math. Comput. 79, 209–237.
  • Andersen, L. (2008). Simple and efficient simulation of the Heston stochastic volatility model. J. Comput. Finance 11, 1–42.
  • Bateman, H. and Erdélyi, A. (1953). Higher Transcendental Functions. McGraw-Hill, New York.
  • Broadie, M. and Kaya, Ö. (2006). Exact simulation of stochastic volatility and other affine jump diffusion processes. Operat. Res. 54, 217–231.
  • 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.
  • 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.
  • Doss, H. (1977). Liens entre équations différentielles stochastiques et ordinaires. Ann. Inst. H. Poincaré Sect. B (N.S.) 13, 99–125.
  • Glasserman, P. (2004). Monte Carlo Methods in Financial Engineering. Springer, New York.
  • Göing-Jaeschke, A. and Yor, M. (2003). A survey on some generalizations of Bessel processes. Bernoulli 9, 313–349.
  • Hartman, P. (1964). Ordinary Differential Equations. John Wiley, New York.
  • 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.
  • Higham, D. J. and Mao, X. (2005). Convergence of Monte Carlo simulations involving the mean-reverting square root process. J. Comp. Finance 8, 35–61.
  • 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.
  • Ikeda, N. and Watanabe, S. (1981). Stochastic Differential Equations and Diffusion Processes. North-Holland, Amsterdam.
  • Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd edn. Springer, New York.
  • Milstein, G. N. and Schoenmakers, J. G. M. (2015). Uniform approximation of the CIR process via exact simulation at random times. Preprint. WIAS preprint no. 2113, ISSN 2198-9855.
  • Milstein, G. N. and Tretyakov, M. V. (2004). Stochastic Numerics for Mathematical Physics. Springer, Berlin.
  • Milstein, G. N. and Tretyakov, M. V. (2005). Numerical analysis of Monte Carlo evaluation of Greeks by finite differences. J. Comp. Finance 8, 1–34.
  • Revuz, D. and Yor, M. (1991). Continuous Martingales and Brownian Motion. Springer, Berlin.
  • Rogers, L. C. G. and Williams, D. (1987). Diffusions, Markov Processes, and Martingales, Vol. 2, Itô Calculus. John Wiley, New York.
  • Sussmann, H. J. (1978). On the gap between deterministic and stochastic ordinary differential equations. Ann. Prob. 6, 19–41.