SpringerLink
Log in

Unconditionally positivity-preserving approximations of the Aït-Sahalia type model: Explicit Milstein-type schemes

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

The present article aims to design and analyze efficient first-order strong schemes for a generalized Aït-Sahalia type model arising in mathematical finance and evolving in a positive domain \((0, \infty )\), which possesses a diffusion term with superlinear growth and a highly nonlinear drift that blows up at the origin. Such a complicated structure of the model unavoidably causes essential difficulties in the construction and convergence analysis of time discretizations. By incorporating implicitness in the term \(\alpha _{-1} x^{-1}\) and a corrective map** \(\Phi _h\) in the recursion, we develop a novel class of explicit and unconditionally positivity-preserving (i.e., for any step-size \(h>0\)) Milstein-type schemes for the underlying model. In both non-critical and general critical cases, we introduce a novel approach to analyze mean-square error bounds of the novel schemes, without relying on a priori high-order moment bounds of the numerical approximations. The expected order-one mean-square convergence is attained for the proposed scheme. The above theoretical guarantee can be used to justify the optimal complexity of the Multilevel Monte Carlo method. Numerical experiments are finally provided to verify the theoretical findings.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Code Availability

No datasets were generated or analysed during the current study.

References

  1. Aït-Sahalia, Y.: Testing continuous-time models of the spot interest rate. The Review of Financial Studies 9(2), 385–426 (1996)

    Article  Google Scholar 

  2. Alfonsi, A.: Strong order one convergence of a drift implicit Euler scheme: Application to the CIR process. Statistics & Probability Letters 83(2), 602–607 (2013)

    Article  MathSciNet  Google Scholar 

  3. Beyn, W., Isaak, E., Kruse, R.: Stochastic C-stability and B-consistency of explicit and implicit Euler-type schemes. J. Sci. Comput. 67, 955–987 (2016)

    Article  MathSciNet  Google Scholar 

  4. Beyn, W., Isaak, E., Kruse, R.: Stochastic C-stability and B-consistency of explicit and implicit Milstein-type schemes. J. Sci. Comput. 70, 1042–1077 (2017)

    Article  MathSciNet  Google Scholar 

  5. Cai, Y., Guo, Q., Mao, X.: Positivity preserving truncated scheme for the stochastic Lotka-Volterra model with small moment convergence. Calcolo 60, 24 (2023)

    Article  MathSciNet  Google Scholar 

  6. Chassagneux, J., Jacquier, A., Mihaylov, I.: An explicit Euler scheme with strong rate of convergence for financial SDEs with non-Lipschitz coefficients. SIAM Journal on Financial Mathematics 7(1), 993–1021 (2016)

    Article  MathSciNet  Google Scholar 

  7. Chen, L., Gan, S., Wang, X.: First order strong convergence of an explicit scheme for the stochastic SIS epidemic model. J. Comput. Appl. Math. 392, 113482 (2021)

    Article  MathSciNet  Google Scholar 

  8. Deng, S., Fei, C., Fei, W., Mao, X.: Positivity-preserving truncated Euler-Maruyama method for generalised Ait-Sahalia-type interest model. BIT Numer. Math. 63(4), 59 (2023)

    Article  MathSciNet  Google Scholar 

  9. Dereich, S., Neuenkirch, A., Szpruch, L.: An Euler-type method for the strong approximation of the Cox-Ingersoll-Ross process. Proceedings of the royal society A: mathematical, physical and engineering sciences 468(2140), 1105–1115 (2012)

    Article  MathSciNet  Google Scholar 

  10. Emmanuel, C., Mao, X.: Truncated EM numerical method for generalised Ait-Sahalia-type interest rate model with delay. J. Comput. Appl. Math. 383, 113137 (2021)

    Article  MathSciNet  Google Scholar 

  11. Halidias, N., Stamatiou, I.S.: Boundary preserving explicit scheme for the Aït-Sahalia model. Discrete and Continuous Dynamical Systems Series B 28(1), 648–664 (2023)

    Article  MathSciNet  Google Scholar 

  12. Higham, D.J., Mao, X., Szpruch, L.: Convergence, non-negativity and stability of a new Milstein scheme with applications to finance. Discrete and Continuous Dynamical Systems Series B 18(8), 2083–2100 (2013)

    Article  MathSciNet  Google Scholar 

  13. Hong, J., Ji, L., Wang, X., Zhang, J.: Positivity-preserving symplectic methods for the stochastic Lotka-Volterra predator-prey model. BIT Numer. Math. 62, 493–520 (2022)

    Article  MathSciNet  Google Scholar 

  14. Hutzenthaler, M., Jentzen, A., Kloeden, P.E.: Strong and weak divergence in finite time of Euler’s method for stochastic differential equations with non-globally Lipschitz continuous coefficients. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 467(2130), 1563–1576 (2011)

    Article  MathSciNet  Google Scholar 

  15. Lei, Z., Gan, S., Chen, Z.: Strong and weak convergence rates of logarithmic transformed truncated EM methods for SDEs with positive solutions. J. Comput. Appl. Math. 419, 114758 (2023)

    Article  MathSciNet  Google Scholar 

  16. Li, Y., Cao, W.: A positivity preserving Lamperti transformed Euler-Maruyama method for solving the stochastic Lotka-Volterra competition model. Commun. Nonlinear Sci. Numer. Simul. 122, 107260 (2023)

    Article  MathSciNet  Google Scholar 

  17. Liu, R., Cao, Y., Wang, X.: Unconditionally positivity-preserving explicit Euler-type schemes for a generalized Aït-Sahalia model. Numerical Algorithms (2024)

  18. Liu, R., Wang, X.: A higher order positivity preserving scheme for the strong approximations of a stochastic epidemic model. Commun. Nonlinear Sci. Numer. Simul. 124, 107258 (2023)

    Article  MathSciNet  Google Scholar 

  19. Lord, G., Wang, M.: Convergence of a exponential tamed method for a general interest rate model. Appl. Math. Comput. 467, 128503 (2024)

    MathSciNet  Google Scholar 

  20. Mao, X., Wei, F., Wiriyakraikul, T.: Positivity preserving truncated Euler-Maruyama Method for stochastic Lotka-Volterra competition model. J. Comput. Appl. Math. 394, 113566 (2021)

    Article  MathSciNet  Google Scholar 

  21. Milstein, G.N., Tretyakov, M.V.: Stochastic Numerics for Mathematical Physics. Springer, Berlin (2004)

    Book  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  23. Platen, E.: An introduction to numerical methods for stochastic differential equations. Acta Numer 8, 197–246 (1999)

    Article  MathSciNet  Google Scholar 

  24. Szpruch, L., Mao, X., Higham, D.J., Pan, J.: Numerical simulation of a strongly nonlinear Ait-Sahalia-type interest rate model. BIT Numer. Math. 51, 405–425 (2011)

    Article  MathSciNet  Google Scholar 

  25. Wang, X.: Mean-square convergence rates of implicit Milstein type methods for SDEs with non-Lipschitz coefficients. Adv. Comput. Math. 49, 37 (2023)

    Article  MathSciNet  Google Scholar 

  26. Wang, X., Wu, J., Dong, B.: Mean-square convergence rates of stochastic theta methods for SDEs under a coupled monotonicity condition. BIT Numer. Math. 60(3), 759–790 (2020)

    Article  MathSciNet  Google Scholar 

  27. Yang, H., Huang, J.: First order strong convergence of positivity preserving logarithmic Euler-Maruyama method for the stochastic SIS epidemic model. Appl. Math. Lett. 121, 107451 (2021)

    Article  MathSciNet  Google Scholar 

  28. Yang, H., Huang, J.: Strong convergence and extinction of positivity preserving explicit scheme for the stochastic SIS epidemic model. Numerical Algorithms 95, 1475–1502 (2024)

    Article  MathSciNet  Google Scholar 

  29. Yi, Y., Hu, Y., Zhao, J.: Positivity preserving logarithmic Euler-Maruyama type scheme for stochastic differential equations. Commun. Nonlinear Sci. Numer. Simul. 101, 105895 (2021)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

Not applicable

Funding

This work is supported by the National Natural Science Foundation of China (Grant Nos. 12071488, 12371417, 11971488), the Natural Science Foundation of Hunan Province (Grant No.2020JJ2040).

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, HNP-LAMA, Central South University, Changsha, Hunan, China

    Yingsong Jiang, Ruishu Liu & **aojie Wang

  2. Department of Economics, University of Warwick, Coventry, CV4 7AL, United Kingdom

    **ghua Zhuo

Authors
  1. Yingsong Jiang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ruishu Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. **aojie Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. **ghua Zhuo
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Yingsong Jiang: Formal analysis, Investigation, Methodology, Writing - Original Draft Ruishu Liu: Formal analysis, Methodology, Validation, Writing - Review & Editing, Software **aojie Wang: Conceptualization, Methodology, Writing - Review & Editing, Funding acquisition, Supervision. **ghua Zhuo: Formal analysis, Investigation, Methodology - Original Draft

Corresponding author

Correspondence to Ruishu Liu.

Ethics declarations

Ethical Approval

Not applicable.

Competing interests

The authors declare no competing interests.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jiang, Y., Liu, R., Wang, X. et al. Unconditionally positivity-preserving approximations of the Aït-Sahalia type model: Explicit Milstein-type schemes. Numer Algor (2024). https://doi.org/10.1007/s11075-024-01861-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11075-024-01861-5

Keywords

Mathematics Subject Classification (2010)

Search

Navigation