Pathwise Taylor schemes for random ordinary differential equations

Article

Abstract

Random ordinary differential equations (RODEs) are ordinary differential equations which contain a stochastic process in their vector fields. They can be analyzed pathwise using deterministic calculus, but since the driving stochastic process is usually only Hölder continuous in time, the vector field is not differentiable in the time variable. Traditional numerical schemes for ordinary differential equations thus do not achieve their usual order of convergence when applied to RODEs. Nevertheless, deterministic calculus can still be used to derive higher order numerical schemes for RODEs by means of a new kind of integral Taylor expansion. The theory is developed systematically here, applied to illustrative examples involving Brownian motion and fractional Brownian motion as the driving processes and compared with other numerical schemes for RODEs in the literature.

Keywords

Random ordinary differential equations Integral Taylor expansion One-step numerical scheme Pathwise convergence Brownian motion Fractional Brownian motion 

Mathematics Subject Classification (2000)

65C30 65L05 65L20 

References

  1. 1.
    Arnold, L.: Random Dynamical Systems. Springer, Heidelberg (1998) MATHGoogle Scholar
  2. 2.
    Bunke, H.: Gewöhnliche Differentialgleichungen mit zufälligen Parametern. Akademie, Berlin (1972) MATHGoogle Scholar
  3. 3.
    Carbonell, F., Jimenez, J.C., Biscay, R.J., de la Cruz, H.: The local linearization method for numerical integration of random differential equations. BIT 45, 1–14 (2005) MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge (1992) MATHGoogle Scholar
  5. 5.
    Deuflhard, P., Bornemann, V.: Scientific Computing with Ordinary Differential Equations. Springer, Berlin (2002) MATHGoogle Scholar
  6. 6.
    Grecksch, W., Kloeden, P.E.: Time-discretised Galerkin approximation of parabolic stochastic PDEs. Bull. Austral. Math. Soc. 54, 79–85 (1996) MathSciNetGoogle Scholar
  7. 7.
    Grüne, L., Kloeden, P.E.: Higher order numerical schemes for affinely controlled nonlinear systems. Numer. Math. 89, 669–690 (2001) MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Grüne, L., Kloeden, P.E.: Pathwise approximation of random ordinary differential equations. BIT 41, 710–721 (2001) CrossRefGoogle Scholar
  9. 9.
    Imkeller, P., Schmalfuß, B.: The conjugacy of stochastic and random differential equations and the existence of global attractors. J. Dyn. Differ. Equ. 13, 215–249 (2001) MATHCrossRefGoogle Scholar
  10. 10.
    Isidori, A.: Nonlinear Control Systems. An Introduction, 2nd edn. Springer, Heidelberg Google Scholar
  11. 11.
    Jentzen, A.: Numerische Verfahren hoher Ordnung für zufällige Differentialgleichungen. Diplomarbeit, J.W. Goethe Universität, Frankfurt am Main, February 2007 Google Scholar
  12. 12.
    Jentzen, A., Kloeden, P.E.: Pathwise convergent higher order numerical schemes for random ordinary differential equations. Proc. R. Soc. Lond. A 463(2087), 2929–2944 (2007) MATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Jentzen, A., Neuenkirch, A.: A random Euler scheme for Carathéodory differential equations. J. Comput. Appl. Math. 224(1), 346–359 (2009) MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Kloeden, P.E., Platen, E.: Numerical Solutions of Stochastic Differential Equations. Springer, Berlin (1992) Google Scholar
  15. 15.
    Prévot, C., Röckner, M.: A Concise Course on Stochastic Partial Differential Equations. Springer, Berlin (2007) MATHGoogle Scholar
  16. 16.
    Sobczyk, K.: Stochastic Differential Equations with Applications to Physics and Engineering. Kluwer Academic, Dordrecht (1991) MATHGoogle Scholar
  17. 17.
    Soong, T.T.: Random Differential Equations in Science and Engineering. Academic Press, San Diego (1973) MATHGoogle Scholar
  18. 18.
    Stengle, G.: Numerical methods for systems with measurable coefficients. Appl. Math. Lett. 3, 25–29 (1990) MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Sussmann, H.J.: On the gap between deterministic and stochastic differential equations. Ann. Probab. 6, 590–603 (1977) MathSciNetGoogle Scholar

Copyright information

© Springer Science + Business Media B.V. 2009

Authors and Affiliations

  1. 1.Department of MathematicsJohann Wolfgang Goethe-UniversityFrankfurt am MainGermany

Personalised recommendations