Skip to main content
SpringerLink
Log in

Asymptotically optimal approximation of some stochastic integrals and its applications to the strong second-order methods

  • Published:
Advances in Computational Mathematics Aims and scope Submit manuscript

Abstract

This study concerns the approximation of some stochastic integrals used in the strong second-order methods for several classes of stochastic differential equations. An explicit construction of the asymptotically optimal approximation (in the mean-square sense) to these stochastic integrals is proposed based on a Karhunen-Loève expansion of a Wiener process. This asymptotically optimal approximation is more efficient by comparison with the Fourier series approximation introduced by Kloeden and Platen (1992) and the Taylor approximation introduced by Milstein and Tretyakov (2004). In the numerical test part, we replace the stochastic integrals appearing in the strong second-order methods with our corresponding approximations. The numerical results show that those strong second-order methods can perform very well by using our approximation method.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Brugnano, L., Burrage, K., Burrage, P.M.: Adams-type methods for the numerical solution of stochastic ordinary differential equations. BIT Numer. Math. 40(3), 451–470 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  2. Buckwar, E., Winkler, R.: Multistep methods for SDEs and their application to problem with small noise. SIAM J. Numer. Anal. 44(2), 779–803 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  3. Buckwar, E., Winkler, R.: Improved linear multi-step methods for stochastic ordinary differential equations. J. Comput. Appl. Math. 205(2), 912–922 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  4. Burrage, K., Burrage, P.M.: High strong order explicit Runge-Kutta methods for stochastic ordinary differential equations. Appl. Numer. Math. 22, 81–101 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  5. Burrage, K., Burrage, P.M.: General order conditions for stochastic Runge-Kutta methods for both commuting and non-commuting stochastic ordinary differential equation systems. Appl. Numer. Math. 28, 161–177 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  6. Burrage, K., Burrage, P.M.: Order conditions of stochastic Runge-Kutta methods by B-series. SIAM J. Numer. Anal. 38(5), 1626–1646 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  7. Burrage, K., Burrage, P.M., Tian, T.: Numerical methods for strong solutions of stochastic differential equations: An overview. Proc. R. Soc. Lond. A 460, 373–402 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cao, W., Zhang, Z.: Simulations of two-step Maruyama methods for nonlinear stochastic delay differential equations. Adv. Appl. Math. Mech. 4(6), 821–832 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. Debrabant, K.: Runge-Kutta methods for third order weak approximation of SDEs with multidimensional additive noise. BIT Numer. Math. 50(3), 541–558 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Debrabant, K., Kværøn, A.: B-series analysis of stochastic Runge-Kutta methods that use an iterative scheme to compute their internal stage values. SIAM J. Numer. Anal. 47(1), 181–203 (2008)

    Article  MathSciNet  Google Scholar 

  11. De la Cruz Cancino, H., Biscay, R.J., Jimenez, J.C., Carbonell, F., Ozaki, T.: High order local linearization methods: An approach for constructing A-stable explicit schemes for stochastic differential equations with additive noise. BIT Numer. Math. 50, 509–539 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  12. Dickinson, A.S.: Optimal approximation of the second iterated integral of Brownian motion. Stoch. Anal. Appl. 25(5), 1109–1128 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  13. Fogelson, A.L.: A mathematical model and numerical method for studying platelet adhesion and aggregation during blood clotting. J. Comput. Phys. 56, 111–134 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gan, S., Henri, S., Zhang, H.: Mean square convergence of stochastic 𝜃-methods for nonlinear neutral stochastic differential delay equations. Int. J. Numer. Anal. Mod. 8, 201–213 (2011)

    MathSciNet  MATH  Google Scholar 

  15. Geman, S., Hwang, C.R.: Diffusions for global optimization. SIAM J. Control Optim. 24(5), 1031–1043 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  16. Gitterman, M.: The noisy oscillator. World Scientific, Singapore (2005)

    Book  MATH  Google Scholar 

  17. Higham, D.J.: An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Rev. 43, 525–546 (2001)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  19. Hong, J., Xu, D., Wang, P.: Preservation of quadratic invariants of stochastic differential equations via Runge-Kutta methods. Appl. Numer. Math. 87, 38–52 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  20. Janson, S.: Gaussian Hilbert spaces. Cambridge University Press, Cambridge (1997)

    Book  MATH  Google Scholar 

  21. Jiménez, J.C., Carbonell, F.: Convergence rate of weak local linearization schemes for stochastic differential equations with additive noise. J. Comput. Appl. Math. 279, 106–122 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  22. Jimenez, J.C., De la Cruz Cancino, H.: Convergence rate of strong local linearization schemes for stochastic differential equations with additive noise. BIT Numer. Math. 52, 357–382 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kamrani, M., Jamshidi, N.: Implicit Milstein method for stochastic differential equations via the Wong-Zakai approximation. Numer. Algor. 79, 357–374 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  24. Kloeden, P.E., Platen, E.: Numerical solution of stochastic differential equations. Springer, Berlin (1992)

    Book  MATH  Google Scholar 

  25. Komori, Y.: Weak second-order stochastic Runge-Kutta methods for non-commutative stochastic differential equations. J. Comput. Appl. Math. 206(1), 158–173 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  26. Komori, Y., Mitsui, T., Sugiura, H.: Rooted tree analysis of the order conditions of row-type scheme for stochastic differential equations. BIT Numer. Math. 37(1), 43–66 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  27. Li, X., Cao, W.: On mean-square stability of two-step Maruyama methods for nonlinear neutral stochastic delay differential equations. Appl. Math. Comput. 261, 373–381 (2015)

    MathSciNet  MATH  Google Scholar 

  28. Mao, X.: Stochastic differential equations and applications. Horword, Chichester (1997)

    MATH  Google Scholar 

  29. Mao, X.: The truncated Euler-Maruyama method for stochastic differential equations. J. Comput. Appl. Math. 290, 370–384 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  30. Mao, X.: Convergence rates of the truncated Euler-Maruyama method for stochastic differential equations. J. Comput. Appl. Math. 296, 362–375 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  31. Milstein, G.N.: Numerical integration of stochastic differential equations. Kluwer, Dordrecht (1995)

    Book  Google Scholar 

  32. Milstein, G.N., Tretyakov, M.V.: Stochastic numerics for mathematical physics. Springer, Berlin (2004)

    Book  MATH  Google Scholar 

  33. Nualart, D.: The Malliavin calculus and related topics. Springer-Verlag, New York (1995)

    Book  MATH  Google Scholar 

  34. Riera, J.J., Wan, X., Jimenez, J.C., Kawashima, R.: Nonlinear local electrovascular coupling. I: a theoretical model. Hum. Brain Mapp. 27, 896–914 (2006)

    Article  Google Scholar 

  35. Rößler, A.: Rooted tree analysis for order conditions of stochastic Runge-Kutta methods for the weak approximation of stochastic differential equations. Stochastic. Anal. Appl. 24(1), 97–134 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  36. Rößler, A.: Second order Runge-Kutta methods for Itô stochastic differential equations. SIAM J. Numer. Anal. 47(3), 1713–1738 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  37. Rößler, A.: Runge-Kutta methods for the strong approximation of solutions of stochastic differential equations. SIAM J. Numer. Anal. 48(3), 922–952 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  38. Tang, X., Xiao, A.: Efficient weak second-order stochastic Runge-Kutta methods for Itô stochastic differential equations. BIT Numer. Math. 57, 241–260 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  39. Wang, X., Gan, S.: A Runge-Kutta type scheme for nonlinear stochastic partial differential equations with multiplicative trace class noise. Numer. Algor. 62 (2), 193–223 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  40. Xiao, A., Tang, X.: High strong order stochastic Runge-Kutta methods for Stratonovich stochastic differential equations with scalar noise. Numer. Algor. 72, 259–296 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  41. Zhou, W., Zhang, J., Hong, J., Song, S.: Stochastic symplectic Runge-Kutta methods for the strong approximation of Hamiltonian systems with additive noise. J. Comput. Appl. Math. 325, 134–148 (2017)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This research is supported by the National Natural Science Foundation of China (No.11671343) and the Hunan Province Innovation Foundation for Postgraduate (No. CX2016B250).

Author information

Authors and Affiliations

  1. School of Mathematics and Computational Science & Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University, Xiangtan, Hunan, 411105, China

    Xiao Tang & Aiguo Xiao

Authors
  1. Xiao Tang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Aiguo Xiao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Aiguo Xiao.

Additional information

Communicated by: Aihui Zhou

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tang, X., Xiao, A. Asymptotically optimal approximation of some stochastic integrals and its applications to the strong second-order methods. Adv Comput Math 45, 813–846 (2019). https://doi.org/10.1007/s10444-018-9638-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10444-018-9638-0

Keywords

Mathematics Subject Classification (2010)

Search

Navigation