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On the backward Euler method for a generalized Ait-Sahalia-type rate model with Poisson jumps

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Abstract

This article aims to reveal the mean-square convergence rate of the backward Euler method (BEM) for a generalized Ait-Sahalia interest rate model with Poisson jumps. The main difficulty in the analysis is caused by the non-globally Lipschitz drift and diffusion coefficients of the model. We show that the BEM preserves the positivity of the original problem. Furthermore, we successfully recover the mean-square convergence rate of order one-half for the BEM. The theoretical findings are accompanied by several numerical examples.

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References

  1. Andersson, A., Kruse, R.: Mean-square convergence of the BDF2-Maruyama and backward Euler schemes for SDE satisfying a global monotonicity condition. BIT Numer. Math. 57, 21–53 (2017)

    Article  MathSciNet  Google Scholar 

  2. Buckwar, E., Riedler, M.: Runge-Kutta methods for jump-diffusion differential equations. J. Comput. Appl. Math. 236, 1155–1182 (2011)

    Article  MathSciNet  Google Scholar 

  3. Chalmers, G., Higham, D.: Asymptotic stability of a jump-diffusion equation and its numerical approximation. SIAM J. Sci. Comput. 31, 1141–1155 (2008)

    Article  MathSciNet  Google Scholar 

  4. Chen, Z., Gan, S.: Convergence and stability of the backward Euler method for jump-diffusion SDEs with super-linearly growing diffusion and jump coefficients. J. Comput. Appl. Math. 365, 350–369 (2020)

    Article  MathSciNet  Google Scholar 

  5. Chen, Z., Gan, S., Wang, X.: Mean-square approximations of Lévy noise driven SDEs with super-linearly growing diffusion and jump coefficients. Discrete Contin. Dynam. Systems-B 24, 4513–4545 (2019)

    Article  Google Scholar 

  6. Cont, R., Tankov, P.: Financial Modelling with Jump Processes. Chapman and Hall/CRC, London (2004)

    MATH  Google Scholar 

  7. Dareiotis, K., Kumar, C., Sabanis, S.: On tamed Euler approximations of SDEs driven by Lévy noise with applications to delay equations. SIAM J. Numer. Anal. 54, 1840–1872 (2016)

    Article  MathSciNet  Google Scholar 

  8. Deng, S., Fei, C., Fei, W., Mao, X.: Generalized Ait-Sahalia-type interest rate model with Poisson jumps and convergence of the numerical approximation. Physica A 533, 122057 (2019)

    Article  MathSciNet  Google Scholar 

  9. Deng, S., Fei, W., Liu, W., Mao, X.: The truncated EM method for stochastic differential equations with Poisson jumps. J. Comput. Appl. Math. 355, 232–257 (2019)

    Article  MathSciNet  Google Scholar 

  10. Gardoń, A.: The order of approximations for solutions of Itô-type stochastic differential equations with jumps. Stoch. Anal. Appl. 22, 679–699 (2004)

    Article  MathSciNet  Google Scholar 

  11. Higham, D., Kloeden, P.: Numerical methods for nonlinear stochastic differential equations with jumps. Numer. Math. 101, 101–119 (2005)

    Article  MathSciNet  Google Scholar 

  12. Higham, D., Kloeden, P.: Convergence and stability of implicit methods for jump diffusion systems. Int. J. Numer. Anal. Model 3, 125–140 (2006)

    MathSciNet  MATH  Google Scholar 

  13. Higham, D., Kloeden, P.: Strong convergence rates for backward Euler on a class of nonlinear jump-diffusion problems. J. Comput. Appl. Math. 205, 949–956 (2007)

    Article  MathSciNet  Google Scholar 

  14. Higham, D., Mao, X., Stuart, A.: Strong convergence of Euler-type methods for nonlinear stochastic differential equations. SIAM J. Numer. Anal. 40, 1041–1063 (2002)

    Article  MathSciNet  Google Scholar 

  15. Huang, C.: Exponential mean square stability of numerical methods for systems of stochastic differential equations. J. Comput. Appl. Math. 236, 4016–4026 (2012)

    Article  MathSciNet  Google Scholar 

  16. Kaluża, A., Przybylowicz, P.: Optimal global approximation of jump-diffusion SDEs via path-independent step-size control. Appl. Numer. Math. 128, 24–42 (2018)

    Article  MathSciNet  Google Scholar 

  17. Kumar, C., Sabanis, S.: On explicit approximations for Lévy driven SDEs with super-linear diffusion coefficients. Electron. J. Probab. 22, 1–19 (2017)

    Article  Google Scholar 

  18. Kumar, C., Sabanis, S.: On tamed Milstein schemes of SDEs driven by Lévy noise. Discrete Contin. Dynam. Systems-B 22, 421–463 (2017)

    Article  Google Scholar 

  19. Li, L., Dai, T.: On a positivity preserving numerical scheme for jump-extended CIR process: the alpha-stable case. BIT Numer. Math. 59, 747–774 (2019)

    Article  MathSciNet  Google Scholar 

  20. Mao, X.: Stochastic Differential Equations and Applications. Horwood (2008)

  21. Mao, X., Szpruch, L.: Strong convergence and stability of implicit numerical methods for stochastic differential equations with non-globally Lipschitz continuous coefficients. J. Comput. Appl. Math. 238, 14–28 (2013)

    Article  MathSciNet  Google Scholar 

  22. Mao, X., Szpruch, L.: Strong convergence rates for backward Euler–Maruyama method for non-linear dissipative-type stochastic differential equations with super-linear diffusion coefficients. Stochastics 85, 144–171 (2013)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  24. Platen, E., Bruti-Liberati, N.: Numerical Solution of Stochastic Differential Equations with Jumps in Finance. Springer Science & Business Media (2010)

  25. Przybylowicz, P.: Optimal global approximation of stochastic differential equations with additive Poisson noise. Numer. Algor. 73, 323–348 (2016)

    Article  MathSciNet  Google Scholar 

  26. Szpruch, L., Mao, X., Higham, D., 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 

  27. Wang, X., Gan, S.: Compensated stochastic theta methods for stochastic differential equations with jumps. Appl. Numer. Math. 60, 877–887 (2010)

    Article  MathSciNet  Google Scholar 

  28. 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, 759–790 (2020)

    Article  MathSciNet  Google Scholar 

  29. Yang, X., Wang, X.: A transformed jump-adapted backward Euler method for jump-extended CIR and CEV models. Numer. Algor. 71, 39–57 (2017)

    Article  MathSciNet  Google Scholar 

  30. Zong, X., Wu, F.: Choice of 𝜃 and mean-square exponential stability in the stochastic theta method of stochastic differential equations. J. Comput. Appl. Math. 255, 837–847 (2014)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics and Statistics, Central South University, Changsha, China

    Yuying Zhao, Xiaojie Wang & Mengchao Wang

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  1. Yuying Zhao
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  2. Xiaojie Wang
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  3. Mengchao Wang
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Correspondence to Mengchao Wang.

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This work was supported by Natural Science Foundation of China (12071488, 11671405, 11971488, 91630312), Innovation Program of Central South University(No.2019zzts397), and Natural Science Foundation of Hunan Province for Distinguished Young Scholars (2020JJ2040).

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Zhao, Y., Wang, X. & Wang, M. On the backward Euler method for a generalized Ait-Sahalia-type rate model with Poisson jumps. Numer Algor 87, 1321–1341 (2021). https://doi.org/10.1007/s11075-020-01009-1

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  • DOI: https://doi.org/10.1007/s11075-020-01009-1

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