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Journal of Probability and Statistics
Volume 2015, Article ID 513137, 20 pages
http://dx.doi.org/10.1155/2015/513137
Research Article

Approximating Explicitly the Mean-Reverting CEV Process

Department of Mathematics, University of the Aegean, Karlovassi, 83 200 Samos, Greece

Received 2 September 2015; Revised 13 October 2015; Accepted 15 October 2015

Academic Editor: Steve Su

Copyright © 2015 N. Halidias and I. S. Stamatiou. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, vol. 23, Springer, Berlin, Germany, 1995, Corrected 2nd printing.
  2. M. J. Brennan and E. S. Schwartz, “Analyzing convertible bonds,” The Journal of Financial and Quantitative Analysis, vol. 15, no. 4, pp. 907–929, 1980. View at Publisher · View at Google Scholar
  3. K. C. Chan, G. A. Karolyi, F. A. Longstaff, and A. B. Sanders, “An empirical comparison of alternative models of the short-term interest rate,” The Journal of Finance, vol. 47, no. 3, pp. 1209–1227, 1992. View at Publisher · View at Google Scholar
  4. A. Alfonsi, “Strong order one convergence of a drift implicit Euler scheme: application to the CIR process,” Statistics & Probability Letters, vol. 83, no. 2, pp. 602–607, 2013. View at Publisher · View at Google Scholar · View at Scopus
  5. N. Halidias, “A new numerical scheme for the CIR process,” Monte Carlo Methods and Applications, vol. 21, no. 3, pp. 245–253, 2015. View at Publisher · View at Google Scholar
  6. J. C. Cox, J. E. Ingersoll, and S. A. Ross, “A theory of the term structure of interest rates,” Econometrica, vol. 53, no. 2, pp. 385–407, 1985. View at Publisher · View at Google Scholar
  7. L. B. G. Andersen and V. V. Piterbarg, “Moment explosions in stochastic volatility models,” Finance and Stochastics, vol. 11, no. 1, pp. 29–50, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  8. N. Halidias, “Semi-discrete approximations for stochastic differential equations and applications,” International Journal of Computer Mathematics, vol. 89, no. 6, pp. 780–794, 2012. View at Publisher · View at Google Scholar · View at Scopus
  9. N. Halidias, “A novel approach to construct numerical methods for stochastic differential equations,” Numerical Algorithms, vol. 66, no. 1, pp. 79–87, 2014. View at Publisher · View at Google Scholar · View at Scopus
  10. T. R. Hurd and A. Kuznetsov, “Explicit formulas for Laplace transforms of stochastic integrals,” Markov Processes and Related Fields, vol. 14, no. 2, pp. 277–290, 2008. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. G. N. Milstein, E. Platen, and H. Schurz, “Balanced implicit methods for stiff stochastic systems,” SIAM Journal on Numerical Analysis, vol. 35, no. 3, pp. 1010–1019, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  12. P. Kloeden and A. Neuenkirch, “Convergence of numerical methods for stochastic differential equations in mathematical finance,” in Recent Developments in Computational Finance: Foundations, Algorithms and Applications, pp. 49–80, 2013. View at Google Scholar
  13. J. Alcock and K. Burrage, “A note on the balanced method,” BIT Numerical Mathematics, vol. 46, no. 4, pp. 689–710, 2006. View at Publisher · View at Google Scholar · View at Scopus
  14. J. Alcock and K. Burrage, “Stable strong order 1.0 schemes for solving stochastic ordinary differential equations,” BIT Numerical Mathematics, vol. 52, no. 3, pp. 539–557, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  15. C. Kahl and P. Jäckel, “Fast strong approximation Monte Carlo schemes for stochastic volatility models,” Quantitative Finance, vol. 6, no. 6, pp. 513–536, 2006. View at Publisher · View at Google Scholar · View at Scopus
  16. N. Halidias, “An explicit and positivity preserving numerical scheme for the mean reverting CEV model,” Japan Journal of Industrial and Applied Mathematics, vol. 32, no. 2, pp. 545–552, 2015. View at Publisher · View at Google Scholar
  17. I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Springer, New York, NY, USA, 1988.
  18. T. H. Gronwall, “Note on the derivatives with respect to a parameter of the solutions of a system of differential equations,” The Annals of Mathematics, vol. 20, no. 4, pp. 292–296, 1919. View at Publisher · View at Google Scholar
  19. T. Yamada and S. Watanabe, “On the uniqueness of solutions of stochastic differential equations,” Journal of Mathematics of Kyoto University, vol. 11, no. 1, pp. 155–167, 1971. View at Google Scholar
  20. X. Mao, Stochastic Differential Equations and Applications, Horwood Publishing, Chichester, UK, 1997.
  21. A. Berkaoui, “Euler scheme for solutions of stochastic differential equations with non-Lipschitz coefficients,” Portugaliae Mathematica, vol. 61, no. 4, pp. 461–478, 2004. View at Google Scholar
  22. S. Dereich, A. Neuenkirch, and L. Szpruch, “An Euler-type method for the strong approximation of the cox-ingersoll-ross process,” Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol. 468, no. 2140, pp. 1105–1115, 2012. View at Publisher · View at Google Scholar
  23. N. Halidias and I. S. Stamatiou, “On the numerical solution of some non-linear stochastic differential equations using the semi-discrete method,” Computational Methods in Applied Mathematics, 2015. View at Publisher · View at Google Scholar
  24. F. W. J. Olver, Asymptotics and Special Functions, AKP Classics, A K Peters, Wellesley, Mass, USA, 1997.
  25. C. Kahl and H. Schurz, “Balanced Milstein methods for ordinary SDEs,” Monte Carlo Methods and Applications, vol. 12, no. 2, pp. 143–170, 2006. View at Publisher · View at Google Scholar · View at Scopus
  26. M. V. Tretyakov and Z. Zhang, “A fundamental mean-square convergence theorem for SDEs with locally Lipschitz coefficients and its applications,” SIAM Journal on Numerical Analysis, vol. 51, no. 6, pp. 3135–3162, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  27. G. Maruyama, “Continuous Markov processes and stochastic equations,” Rendiconti del Circolo Matematico di Palermo, vol. 4, no. 1, pp. 48–90, 1955. View at Publisher · View at Google Scholar · View at Scopus
  28. P. E. Kloeden, E. Platen, and H. Schurz, Numerical Solution of SDE through Computer Experiments, Springer, 2003.
  29. H. Schurz, “Numerical regularization for SDEs: construction of nonnegative solutions,” Dynamic Systems and Applications, vol. 5, no. 3, pp. 323–352, 1996. View at Google Scholar · View at MathSciNet