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Mean-square convergence of numerical methods for random ordinary differential equations

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Abstract

The aim of this work is to analyze the mean-square convergence rates of numerical schemes for random ordinary differential equations (RODEs). First, a relation between the global and local mean-square convergence order of one-step explicit approximations is established. Then, the global mean-square convergence rates are investigated for RODE-Taylor schemes for general RODEs, Affine-RODE-Taylor schemes for RODEs with affine noise, and Itô-Taylor schemes for RODEs with Itô noise, respectively. The theoretical convergence results are demonstrated through numerical experiments.

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References

  1. Asai, Y., Kloeden, P.E.: Numerical schemes for random ODES via stochastic differential equations. Commun. Appl. Anal. 17, 511–528 (2013)

    MathSciNet  MATH  Google Scholar 

  2. Asai, Y., Kloeden, P.E.: Multi-step methods for random ODES driven by Itô diffusions. J. Comput. Appl. Math. 294, 210–224 (2016)

    Article  MathSciNet  Google Scholar 

  3. Asai, Y., Kloeden, P.E.: Numerical schemes for random ODES with affine noise. Numer. Algorithms 72, 155–171 (2016)

    Article  MathSciNet  Google Scholar 

  4. Asai, Y., Herrmann, E., Kloeden, P.E.: Stable integration of stiff random ordinary differential equations. Stoch. Anal. Appl. 31, 293–313 (2013)

    Article  MathSciNet  Google Scholar 

  5. Bunke, H.: Gewöhnliche Differentialgleichungen mit zufälligen Parametern. Akademie-Verlag, Berlin (1972)

    MATH  Google Scholar 

  6. Chen, C., Hong, J.: Mean-square convergence of numerical approximations for a class of backward stochastic differential equations. Discrete Contin. Dyn. Syst. Ser. B 8, 2051–2067 (2013)

    MathSciNet  MATH  Google Scholar 

  7. Cortes, J., Jordan, L., Villafuerte, L.: Numerical solution of random differential equations, a mean square approach. Math. Comput. Model. 45, 757–765 (2007)

    Article  MathSciNet  Google Scholar 

  8. Grüne, L., Kloeden, P.E.: Pathwise approximation of random ordinary differential equations. BIT 41, 711–721 (2001)

    Article  MathSciNet  Google Scholar 

  9. Han, X., Kloeden, P.E.: Random Ordinary Differential Equations and Their Numerical Solution. Springer Nature, Singapore (2017)

    Book  Google Scholar 

  10. Jentzen, A., Kloeden, P.E.: Pathwise taylor schemes for random ordinary differential equations. BIT Numer. Math. 49, 113–140 (2009)

    Article  MathSciNet  Google Scholar 

  11. Jentzen, A., Kloeden, P.E.: Taylor Approximations of Stochastic Partial Differential Equations. SIAM, Philadelphia (2011)

    Book  Google Scholar 

  12. Kloeden, P.E., Jentzen, A.: Pathwise convergent higher order numerical schemes for random ordinary. Proc. R. Soc. A 463, 2929–2944 (2007)

    Article  MathSciNet  Google Scholar 

  13. Kloeden, P.E., Platen, E.: Numerical Solution of Stochastic Differential Equations. Springer, Heidelberg (1992)

    Book  Google Scholar 

  14. Milstein, G.N.: Numerical Integration of Stochastic Differential Equations. Kluwer Acdemic Publishers Group, Dordrecht (1995)

    Book  Google Scholar 

  15. Milstein, G.N.: Tretyakov: Stochastic Numerics for Mathematical Physics. Springer, New York (2004)

    Book  Google Scholar 

  16. Stuart, A.M., Humphries, A.R.: Dynamical Systems and Numerical Analysis. Cambridge University Press, Cambridge (1996)

    MATH  Google Scholar 

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Acknowledgments

The authors would like to express their sincere gratitude to Prof. Dr. Arnulf Jentzen for his insightful comments. The authors would also like to thank the anonymous referees for their insightful comments, which have led to the much improved quality of this work.

Funding

Wang is partially supported by the National Natural Science Foundation of China (11101184, 11271151), Scientific and Technological Development Plan of Jilin Province (20200201251JC) and China Scholarship Council (201506175081), Cao is partially supported by the National Science Foundation (DMS1620150), Han is partially supported by Simons Foundation (Collaboration Grants for Mathematicians 429717), and Kloeden is partially supported by the National Natural Science Foundation of China (11571125).

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Authors and Affiliations

  1. Institute of Mathematics, Jilin University, Changchun, 130012, People’s Republic of China

    Peng Wang

  2. Department of Mathematics and Statistics, Auburn University, Auburn, AL, 36849, USA

    Yanzhao Cao, Xiaoying Han & Peter Kloeden

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  1. Peng Wang
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  2. Yanzhao Cao
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  3. Xiaoying Han
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  4. Peter Kloeden
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Correspondence to Xiaoying Han.

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Wang, P., Cao, Y., Han, X. et al. Mean-square convergence of numerical methods for random ordinary differential equations. Numer Algor 87, 299–333 (2021). https://doi.org/10.1007/s11075-020-00967-w

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  • DOI: https://doi.org/10.1007/s11075-020-00967-w

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