photo Arnulf Jentzen
Princeton University
Address:
Prof. Dr. Arnulf Jentzen
Program in Applied and Computational Mathematics
Princeton University
Fine Hall, Washington Road
Princeton, NJ 08544-1000
USA

Office: Room 219A
Fon: +1-609-258 2654
Fax: +1-609-258 1735

 E-mail: ajentzen (at) math.princeton.edu

Research Profile

Stochastic differential equations (SDEs) by which we mean both stochastic ordinary differential equations (SODEs) and stochastic partial differential equations (SPDEs) are a fundamental instrument for modeling processes with uncertainties in nature and in man-made complex systems. Since explicit solutions of such equations are typically not available, it is a very active topic of research in the last four decades to solve SDEs approximatively. At the moment there is, however, a big discrepancy between the assumptions used for numerical approximation methods for SDEs and the assumptions fulfilled by such equations in "real world" applications, e.g., in population dynamics, in molecular dynamics or in option pricing. More precisely, in many of such applications, SDEs with non-globally Lipschitz nonlinearities appear while in the vast majority of research articles for approximating SDEs the nonlinear terms in the SDE are assumed to be globally Lipschitz continuous. In particular, it has been an open question whether the standard numerical method for approximating SODEs, i.e. the stochastic Euler scheme, converges strongly to the exact solution of an SODE with a superlinearly growing (and hence not globally Lipschitz continuous) drift coefficient such as a cubic drift of the form x-x^3. This problem is precisely described on page 1060 in [1]. The recent article [2] answers this question to the negative and shows that the stochastic Euler scheme fails to converge strongly to the exact solution of such an SODE. Even worse, Theorem 2.1 in [2] establishes that the strong mean square distance of the exact solution and of the stochastic Euler approximation diverges to infinity. Starting from this recent development for SODEs, my research goal is to construct and to analyze new algorithms which overcome the lack of strong convergence of Euler's method and which solve SODEs and SPDEs with non-globally Lipschitz continuous nonlinearities approximatively.

References

  1. Higham, D. J., Mao, X. and Stuart, A. M.,
    Strong convergence of Euler type methods for nonlinear stochastic differential equations.
    SIAM Journal on Numerical Analysis 40 (2002), no. 3, 1041-1063.
  2. Hutzenthaler, M., Jentzen, A. and Kloeden, P. E.,
    Strong and weak divergence in finite time of Euler's method for stochastic differential equations with non-globally Lipschitz continuous coefficients.
    Proceedings of the Royal Society A 467 (2011), no. 2130, 1563-1576. [arXiv].

Publications and Preprints

Preprints
  • Hutzenthaler, M. and Jentzen, A.,
    Numerical approximations of stochastic differential equations with non-globally Lipschitz continuous coefficients.
    [arXiv] (2012), 59 pages.
  • Jentzen, A. and Röckner, M.,
    A Milstein scheme for SPDEs.
    [arXiv] (2012), 37 pages.
  • Hutzenthaler, M., Jentzen, A. and Kloeden, P. E.,
    Divergence of the multilevel Monte Carlo Euler method for nonlinear stochastic differential equations.
    [arXiv] (2011), 31 pages.
  • Da Prato, G., Jentzen, A. and Röckner, M.,
    A mild Ito formula for SPDEs.
    [arXiv] (2011), 31 pages.
  • Blömker, D. and Jentzen, A.,
    Galerkin approximations for the stochastic Burgers equation.
    OPUS Augsburg (2009), 46 pages.
Articles in refereed journals
  • Hutzenthaler, M., Jentzen, A. and Kloeden, P. E.,
    Strong convergence of an explicit numerical method for SDEs with non-globally Lipschitz continuous coefficients.
    To appear in The Annals of Applied Probability (2012). [arXiv].
  • Jentzen, A. and Röckner, M.,
    Regularity analysis for stochastic partial differential equations with nonlinear multiplicative trace class noise.
    Journal of Differential Equations 252 (2012), no. 1, 114-136. [arXiv].
  • Hutzenthaler, M. and Jentzen, A.,
    Convergence of the stochastic Euler scheme for locally Lipschitz Coefficients.
    Foundations of Computational Mathematics 11 (2011), no. 6, 657-706. [arXiv].
  • Hutzenthaler, M., Jentzen, A. and Kloeden, P. E.,
    Strong and weak divergence in finite time of Euler's method for stochastic differential equations with non-globally Lipschitz continuous coefficients.
    Proceedings of the Royal Society A 467 (2011), no. 2130, 1563-1576. [arXiv].
  • Jentzen, A.,
    Higher order pathwise numerical approximations of SPDEs with additive noise.
    SIAM Journal on Numerical Analysis 49 (2011), no. 2, 642-667.
  • Jentzen, A., Kloeden, P. E. and Winkel, G.,
    Efficient simulation of nonlinear parabolic SPDEs with additive noise.
    The Annals of Applied Probability 21 (2011), no. 3, 908-950.
  • Jentzen, A.,
    Taylor expansions of solutions of stochastic partial differential equations.
    Discrete and Continuous Dynamical Systems B 14 (2010), no. 2, 515-557. [arXiv].
  • Jentzen, A. and Kloeden, P. E.,
    Taylor expansions of solutions of stochastic partial differential equations with additive noise.
    The Annals of Probability 38 (2010), no. 2, 532-569. [arXiv].
  • Jentzen, A. and Kloeden, P. E.,
    A unified existence and uniqueness theorem for stochastic evolution equations.
    Bulletin of the Australian Mathematical Society 81 (2010), no. 1, 33-46.
  • Jentzen, A., Leber, F., Schneisgen, D., Berger, A. and Siegmund., S.,
    An improved maximum allowable transfer interval for Lp-stability of networked control systems.
    IEEE Transactions on Automatic Control 55 (2010), no. 1, 179-184.
  • Jentzen, A. and Kloeden, P. E.,
    The numerical approximation of stochastic partial differential equations.
    Milan Journal of Mathematics 77 (2009), no. 1, 205-244.
  • Jentzen, A.,
    Pathwise numerical approximations of SPDEs with additive noise under non-global Lipschitz coefficients.
    Potential Analysis 31 (2009), no. 4, 375-404.
  • Jentzen, A. and Kloeden, P. E.,
    Overcoming the order barrier in the numerical approximation of stochastic partial differential equations with additive space-time noise.
    Proceedings of the Royal Society A 465 (2009), no. 2102, 649-667.
  • Jentzen, A. and Neuenkirch, A.,
    A random Euler scheme for Carathéodory differential equations.
    Journal of Computational and Applied Mathematics 224 (2009), no. 1, 346-359.
  • Jentzen, A. and Kloeden, P. E.,
    Pathwise Taylor schemes for random ordinary differential equations.
    BIT Numerical Mathematics 49 (2009), no. 1, 113-140.
  • Jentzen, A., Kloeden, P. E. and Neuenkirch, A.,
    Pathwise approximation of stochastic differential equations on domains: higher order convergence rates without global Lipschitz coefficients.
    Numerische Mathematik 112 (2009), no. 1, 41-64.
  • Jentzen, A. and Kloeden, P. E.,
    Pathwise convergent higher order numerical schemes for random ordinary differential equations.
    Proceedings of the Royal Society A 463 (2007), no. 2087, 2929-2944.
Conference proceedings, chapters in books, etc.
  • Chapter 5 and Appendix of:
    Jentzen, A. and Kloeden, P. E.,
    Taylor Approximations for Stochastic Partial Differential Equations.
    CBMS-NSF Regional Conference Series in Applied Mathematics 83, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. xiv+211 pp.
  • Jentzen, A., Kloeden, P. E. and Neuenkirch, A.,
    Pathwise convergence of numerical schemes for random and stochastic differential equations.
    Foundations of Computational Mathematics, Hong Kong 2008, 140-161, London Mathematical Society Lecture Note Series, 363, Cambridge University Press, Cambridge, 2009.
Theses
  • Jentzen, A.,
    Taylor Expansions for Stochastic Partial Differential Equations.
    PhD thesis (2009), Johann Wolfgang Goethe University Frankfurt am Main.
  • Jentzen, A.,
    Numerische Verfahren hoher Ordnung für zufällige Differentialgleichungen.
    Diploma thesis (2007), Johann Wolfgang Goethe University Frankfurt am Main.