Implicit Milstein Schemes: Preservation of Properties When Solving the CIR Equation

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Advances in Nonlinear Dynamics, Volume I (ICNDA 2023)

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References

  1. Alfonsi, A.: On the discretization schemes for the CIR (and Bessel squared) processes. Monte Carlo Methods Appl. 11, 355–384 (2005)

    Article  MathSciNet  Google Scholar 

  2. Cox, J., Ingersoll, J., Ross, S.: A Theory of the Term Structure of Interest Rates. World Scientific, Singapore (2005), pp. 129–164

    Google Scholar 

  3. Dereich, S., Neuenkirch, A., Szpruch, L.: An Euler-type method for the strong approximation of the Cox–Ingersoll–Ross process. Proc. R. Soc. A. 468, 1105–1115 (2012)

    Article  ADS  MathSciNet  Google Scholar 

  4. Gikhman, I.I.: A Short Remark on Feller’s Square Root Condition (2011)

    Google Scholar 

  5. Hefter, M., Herzwurm, A.: Strong convergence rates for Cox–Ingersoll–Ross processes–full parameter range. J. Math. Anal. Appl. 459(2), 1079–1101 (2018)

    Article  MathSciNet  Google Scholar 

  6. Higham, D.: A-stability and stochastic mean-square stability. BIT 40, 404–409 (2000)

    Article  MathSciNet  Google Scholar 

  7. Higham, D., Mao, X.: Convergence of Monte Carlo simulations involving the mean-reverting square root process. J. Comput. Financ. 8(3), 35–61 (2005)

    Article  Google Scholar 

  8. Kahl, C., Günther, M., Rossberg, T.: Structure preserving stochastic integration schemes in interest rate derivative modeling. Appl. Numer. Math. 58(3), 284–295 (2008)

    Article  MathSciNet  Google Scholar 

  9. Kladívko, K.: Maximum likelihood estimation of the Cox-Ingersoll-Ross process: the Matlab implementation. Technical Computing Prague 7(8), 1–8 (2007)

    Google Scholar 

  10. Kloeden, P., Platen, E.: Stochastic Differential Equations. Springer, Heidelberg (1992)

    Google Scholar 

  11. Llamazares-Elias, S., Tocino, A.: Mean-Reverting Schemes for Solving the CIR Model. Submitted

    Google Scholar 

  12. Mao, X.: Stochastic Differential Equations and Applications. Elsevier, Amsterdam (2007)

    Google Scholar 

  13. Milstein, G.N.: Numerical Integration of Stochastic Differential Equations. Kluwer Academic Publishers, Amsterdam (1995)

    Book  Google Scholar 

  14. Neuenkirch, A., Szpruch, L.: First order strong approximations of scalar SDEs with values in a domain. Numer. Math. 128(1), 103–136 (2014)

    Article  MathSciNet  Google Scholar 

  15. Øksendal, B.: Stochastic Differential Equations. Springer, New York (2003)

    Book  Google Scholar 

  16. Scalone, C.: Positivity preserving stochastic \(\theta \)-methods for selected SDEs. Appl. Numer. Math. 172, 351–358 (2022)

    Google Scholar 

  17. Shreve, S.E.: Stochastic Calculus for Finance II: Continuous-time Models. Springer, Berlin (2004)

    Book  Google Scholar 

  18. Yamada, Y., Watanabe, S.: On the uniqueness of solutions of stochastic differential equations. J. Math. Kyoto Univ. 11, 155–167 (1971)

    MathSciNet  Google Scholar 

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Correspondence to Angel Tocino .

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Tocino, A., Llamazares-Elias, S. (2024). Implicit Milstein Schemes: Preservation of Properties When Solving the CIR Equation. In: Lacarbonara, W. (eds) Advances in Nonlinear Dynamics, Volume I. ICNDA 2023. NODYCON Conference Proceedings Series. Springer, Cham. https://doi.org/10.1007/978-3-031-50631-4_39

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