Advertisement

Foundations of Computational Mathematics

, Volume 11, Issue 6, pp 657–706 | Cite as

Convergence of the Stochastic Euler Scheme for Locally Lipschitz Coefficients

  • Martin Hutzenthaler
  • Arnulf Jentzen
Article

Abstract

Stochastic differential equations are often simulated with the Monte Carlo Euler method. Convergence of this method is well understood in the case of globally Lipschitz continuous coefficients of the stochastic differential equation. However, the important case of superlinearly growing coefficients has remained an open question. The main difficulty is that numerically weak convergence fails to hold in many cases of superlinearly growing coefficients. In this paper we overcome this difficulty and establish convergence of the Monte Carlo Euler method for a large class of one-dimensional stochastic differential equations whose drift functions have at most polynomial growth.

Keywords

Euler scheme Stochastic differential equations Monte Carlo Euler method Local Lipschitz condition 

Mathematics Subject Classification (2000)

65C05 60H35 65C30 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    V. Bally, D. Talay, The law of the Euler scheme for stochastic differential equations. I. Convergence rate of the distribution function, Probab. Theory Relat. Fields 104(1), 43–60 (1996). MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    D. Duffie, P. Glynn, Efficient Monte Carlo simulation of security prices, Ann. Appl. Probab. 5(4), 897–905 (1995). MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Ĭ.Ī. Gīhman, A.V. Skorohod, Stochastic Differential Equations (Springer, New York, 1972). Translated from the Russian by Kenneth Wickwire, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 72. zbMATHGoogle Scholar
  4. 4.
    P. Glasserman, Monte Carlo Methods in Financial Engineering (Springer, New York, 2004). zbMATHGoogle Scholar
  5. 5.
    D.J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev. 43(3), 525–546 (2001) (electronic). MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    D.J. Higham, X. Mao, A.M. Stuart, Strong convergence of Euler-type methods for nonlinear stochastic differential equations, SIAM J. Numer. Anal. 40(3), 1041–1063 (2002) (electronic). MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    M. Hutzenthaler, A. Jentzen, P.E. Kloeden, Strong and weak divergence in finite time of Euler’s method for stochastic differential equations with non-globally Lipschitz continuous coefficients, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 467(2130), 1563–1576 (2011). MathSciNetCrossRefGoogle Scholar
  8. 8.
    A. Klenke, Probability Theory, Universitext (Springer, London, 2008). A comprehensive course. Translated from the 2006 German original. CrossRefGoogle Scholar
  9. 9.
    P.E. Kloeden, The systematic derivation of higher order numerical schemes for stochastic differential equations, Milan J. Math. 70, 187–207 (2002). MathSciNetCrossRefGoogle Scholar
  10. 10.
    P.E. Kloeden, A. Neuenkirch, The pathwise convergence of approximation schemes for stochastic differential equations. LMS J. Comput. Math. 10, 235–253 (2007) MathSciNetzbMATHGoogle Scholar
  11. 11.
    P.E. Kloeden, E. Platen, Numerical Solution of Stochastic Differential Equations (Springer, Berlin, 1992). zbMATHGoogle Scholar
  12. 12.
    P.E. Kloeden, E. Platen, H. Schurz, Numerical Solution of SDE Through Computer Experiments, Universitext (Springer, Berlin, 1994). zbMATHCrossRefGoogle Scholar
  13. 13.
    J.C. Mattingly, A.M. Stuart, D.J. Higham, Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise, Stoch. Process. Appl. 101(2), 185–232 (2002). MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    G.N. Milstein, A method with second order accuracy for the integration of stochastic differential equations, Teor. Veroâtn. Ee Primen. 23(2), 414–419 (1978). MathSciNetGoogle Scholar
  15. 15.
    G.N. Milstein, Numerical Integration of Stochastic Differential Equations (Kluwer Academic, Dordrecht, 1995). Google Scholar
  16. 16.
    G.N. Milstein, M.V. Tretyakov, Numerical integration of stochastic differential equations with nonglobally Lipschitz coefficients, SIAM J. Numer. Anal. 43(3), 1139–1154 (2005) (electronic). MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    A. Rößler, Runge–Kutta Methods for the Numerical Solution of Stochastic Differential Equations (Shaker Verlag, Aachen, 2003). zbMATHGoogle Scholar
  18. 18.
    D.W. Stroock, Probability Theory, an Analytic View (Cambridge University Press, London, 1993). zbMATHGoogle Scholar
  19. 19.
    L. Szpruch, Numerical approximations of nonlinear stochastic systems, Ph.D. thesis, University of Strathclyde, Glasgow, 2010. Google Scholar
  20. 20.
    D. Talay, Probabilistic numerical methods for partial differential equations: elements of analysis, in Probabilistic Models for Nonlinear Partial Differential Equations (Springer, Berlin, 1996), pp. 148–196. CrossRefGoogle Scholar
  21. 21.
    L. Yan, The Euler scheme with irregular coefficients, Ann. Probab. 30(3), 1172–1194 (2002). MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© SFoCM 2011

Authors and Affiliations

  1. 1.LMU Biozentrum, Department Biologie IIUniversity of Munich (LMU)Planegg-MartinsriedGermany
  2. 2.Program in Applied and Computational MathematicsPrinceton UniversityPrincetonUSA

Personalised recommendations