photo Arnulf Jentzen
University of Münster

Address:
Prof. Dr. Arnulf Jentzen
Institute for Analysis and Numerics
Applied Mathematics Münster
Faculty of Mathematics and Computer Science
University of Münster
Orleansring 10
48149 Münster
Germany

Office: Room 120.005
Fon (Secretariat): +49 251 83-33792
Fax: +49 251 83-32729
Office hour: on appointment

E-mail: ajentzen (at) uni-muenster.de
Homepage: http://www.ajentzen.de
Homepage at the University of Münster: https://www.uni-muenster.de/AMM/show_perspage.shtml?id=1406
Born: November 1983 (age 36)

Links: [Profile on Google Scholar] [Profile on ResearchGate] [Profile on MathSciNet] [ETH Webmail] [Stochastic Computation Workshop 2017]
Last update of this homepage: January 14th, 2020

Research group

Current members of the research group

  • Christian Beck (PhD student at ETH Zurich, D-MATH, joint supervision with Prof. Dr. Norbert Hungerbühler)
  • Prof. Dr. Arnulf Jentzen (Head of the research group)
  • Dr. Benno Kuckuck (Postdoc at the Faculty of Mathematics and Computer Science, University of Muenster)
  • Primoz Pusnik (PhD Student at ETH Zurich, D-MATH, Seminar for Applied Mathematics)
  • Florian Rossmannek (PhD student at ETH Zurich, D-MATH, joint supervision with Prof. Dr. Patrick Cheridito)
  • Philippe von Wurstemberger (PhD student at ETH Zurich, D-MATH, joint supervision with Prof. Dr. Patrick Cheridito)
  • Timo Welti (PhD Student at ETH Zurich, D-MATH, Seminar for Applied Mathematics)

Former members of the research group

  • Dr. Sebastian Becker (former PhD student, joint supervision with Prof. Dr. Peter E. Kloeden, 2010-2017, now Head of Machine Learning Research at Zenai AG)
  • Prof. Dr. Sonja Cox (former Postdoc/Fellow, 2012-2014, now tenured Associate Professor at the University of Amsterdam)
  • Dr. Fabian Hornung (former Postdoc/Fellow, 2018-2018, now Postdoc at the Karlsruhe Institute of Technology)
  • Dr. Raphael Kruse (former Postdoc, 2012-2014, now Head of the Junior Reseach Group "Uncertainty Quantification" at the Technical University of Berlin)
  • Dr. Ryan Kurniawan (former PhD student, 2014-2018, now Associate at Market Risk Analytics at Morgan Stanley UK Ltd.)
  • Prof. Dr. Ariel Neufeld (former Postdoc/Fellow, joint mentoring with Prof. Dr. Patrick Cheridito, 2018-2018, now tenure-track Assistant Professor at NTU Singapore)
  • Dr. Diyora Salimova (former PhD student, 2015-2019, now Postdoc at ETH Zurich, D-MATH, Seminar for Applied Mathematics)
  • Prof. Dr. Michaela Szoelgyenyi (former Postdoc/Fellow, 2017-2018, now full professor at the University of Klagenfurt)
  • Dr. Larisa Yaroslavtseva (former Postdoc, 2018-2018, now Lecturer at the University of Passau)

Research areas

  • Machine learning (mathematics for deep learning, stochastic gradient descent methods, deep artificial neural networks, empirical risk minimization)
  • Stochastic analysis (stochastic calculus, well-posedness and regularity analysis for stochastic ordinary and partial differential equations)
  • Numerical analysis (computational stochastics/stochastic numerics, computational finance)
  • Analysis of partial differential equations (well-posedness and regularity analysis for partial differential equations)

Current editorial boards affiliations

Past editorial boards affiliations

Preprints and accepted publications

  • Jentzen, A., Kuckuck, B., Mueller-Gronbach, T., Yaroslavtseva, L., Counterexamples to local Lipschitz and local Hölder continuity with respect to the initial values for additive noise driven SDEs with smooth drift coefficient functions with at most polynomially growing derivatives. [arXiv] (2020), 27 pages.
  • Becker, S., Cheridito, P., Jentzen, A., Pricing and hedging American-style options with deep learning. [arXiv] (2019), 12 pages.
  • Cheridito, P., Jentzen, A., Rossmannek, F., Efficient approximation of high-dimensional functions with deep neural networks. [arXiv] (2019), 19 pages.
  • Hutzenthaler, M., Jentzen, A., Kruse, T., Overcoming the curse of dimensionality in the numerical approximation of parabolic partial differential equations with gradient-dependent nonlinearities. [arXiv] (2019), 33 pages.
  • Gonon, L., Grohs, P., Jentzen, A., Kofler, D., Siska, D., Uniform error estimates for artificial neural network approximations for heat equations. [arXiv] (2019), 70 pages.
  • Giles, M. B., Jentzen, A., Welti, T., Generalised multilevel Picard approximations. [arXiv] (2019), 61 pages.
  • Hutzenthaler, M., Jentzen, A., Lindner, F., Pusnik, P., Strong convergence rates on the whole probability space for space-time discrete numerical approximation schemes for stochastic Burgers equations. [arXiv] (2019), 60 pages.
  • Beck, C., Jentzen, A., Kuckuck, B., Full error analysis for the training of deep neural networks. [arXiv] (2019), 43 pages.
  • Grohs, P., Jentzen, A., Salimova, D., Deep neural network approximations for Monte Carlo algorithms. [arXiv] (2019), 45 pages.
  • Jentzen, A., Lindner, F., Pusnik, P., Spatial Sobolev regularity for stochastic Burgers equations with additive trace class noise. [arXiv] (2019), 54 pages.
  • Grohs, P., Hornung, F., Jentzen, A., Zimmermann, P., Space-time error estimates for deep neural network approximations for differential equations. [arXiv] (2019), 86 pages.
  • Beck, C., Gonon, L., Hutzenthaler, M., Jentzen, A., On existence and uniqueness properties for solutions of stochastic fixed point equations. [arXiv] (2019), 33 pages.
  • Becker, S., Cheridito, P., Jentzen, A., Welti, T., Solving high-dimensional optimal stopping problems using deep learning. [arXiv] (2019), 42 pages.
  • Beck, C., Hornung, F., Hutzenthaler, M., Jentzen, A., Kruse, T., Overcoming the curse of dimensionality in the numerical approximation of Allen-Cahn partial differential equations via truncated full-history recursive multilevel Picard approximations. [arXiv] (2019), 30 pages.
  • Beck, C., Becker, S., Cheridito, P., Jentzen, A., Neufeld, A., Deep splitting method for parabolic PDEs. [arXiv] (2019), 40 pages.
  • Berner, J., Elbraechter, D., Grohs, P., Jentzen, A., Towards a regularity theory for ReLU networks -- chain rule and global error estimates. [arXiv] (2019), 5 pages.
  • Jentzen, A., Kuckuck, B., Mueller-Gronbach, T., Yaroslavtseva, L., On the strong regularity of degenerate additive noise driven stochastic differential equations with respect to their initial values. [arXiv] (2019), 59 pages.
  • Fehrman, B., Gess, B., Jentzen, A., Convergence rates for the stochastic gradient descent method for non-convex objective functions. [arXiv] (2019), 52 pages.
  • Beccari, M., Hutzenthaler, M., Jentzen, A., Kurniawan, R., Lindner, F., Salimova, D., Strong and weak divergence of exponential and linear-implicit Euler approximations for stochastic partial differential equations with superlinearly growing nonlinearities. [arXiv] (2019), 65 pages.
  • Hutzenthaler, M., Jentzen, A., von Wurstemberger, P., Overcoming the curse of dimensionality in the approximative pricing of financial derivatives with default risks. [arXiv] (2019), 71 pages.
  • Hutzenthaler, M., Jentzen, A., Kruse, T., Nguyen, T. A., A proof that rectified deep neural networks overcome the curse of dimensionality in the numerical approximation of semilinear heat equations. [arXiv] (2019), 24 pages.
  • Cox, S., Jentzen, A., Lindner, F., Weak convergence rates for temporal numerical approximations of stochastic wave equations with multiplicative noise. [arXiv] (2019), 51 pages.
  • Hudde, A., Hutzenthaler, M., Jentzen, A., Mazzonetto, S., On the Itô-Alekseev-Gröbner formula for stochastic differential equations. [arXiv] (2018), 29 pages.
  • Jentzen, A., Lindner, F., Pusnik, P., Exponential moment bounds and strong convergence rates for tamed-truncated numerical approximations of stochastic convolutions. [arXiv] (2018), 25 pages. Major revision requested from Numerical Algorithms.
  • Becker, S., Gess, B., Jentzen, A., Kloeden, P. E., Lower and upper bounds for strong approximation errors for numerical approximations of stochastic heat equations. [arXiv] (2018), 20 pages.
  • Elbraechter, D., Grohs, P., Jentzen, A., Schwab, C., DNN Expression Rate Analysis of High-dimensional PDEs: Application to Option Pricing. [arXiv] (2018), 50 pages.
  • Jentzen, A., Salimova, D., Welti, T., A proof that deep artificial neural networks overcome the curse of dimensionality in the numerical approximation of Kolmogorov partial differential equations with constant diffusion and nonlinear drift coefficients. [arXiv] (2018), 48 pages.
  • Berner, J., Grohs, P., Jentzen, A., Analysis of the generalization error: Empirical risk minimization over deep artificial neural networks overcomes the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations. [arXiv] (2018), 35 pages.
  • Grohs, P., Hornung, F., Jentzen, A., von Wurstemberger, P., A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations. [arXiv] (2018), 124 pages. Revision requested from the Memoires of the American Mathematical Society.
  • Hutzenthaler, M., Jentzen, A., Kruse, T., Nguyen, T. A., von Wurstemberger, P., Overcoming the curse of dimensionality in the numerical approximation of semilinear parabolic partial differential equations. [arXiv] (2018), 30 pages.
  • Beck, C., Becker, S., Grohs, P., Jaafari, N., Jentzen, A., Solving stochastic differential equations and Kolmogorov equations by means of deep learning. [arXiv] (2018), 56 pages.
  • Jentzen, A., Kuckuck, B., Neufeld, A., von Wurstemberger, P., Strong error analysis for stochastic gradient descent optimization algorithms. [arXiv] (2018), 75 pages.
  • Becker, S., Gess, B., Jentzen, A., Kloeden, P. E., Strong convergence rates for explicit space-time discrete numerical approximations of stochastic Allen-Cahn equations. [arXiv] (2017), 104 pages.
  • Hefter, M., Jentzen, A., Kurniawan, R., Counterexamples to regularities for the derivative processes associated to stochastic evolution equations. [arXiv] (2017), 26 pages.
  • E, W., Hutzenthaler, M., Jentzen, A., and Kruse, T., Linear scaling algorithms for solving high-dimensional nonlinear parabolic differential equations. [arXiv] (2017), 18 pages.
  • Hefter, M., Jentzen, A., and Kurniawan, R., Weak convergence rates for numerical approximations of stochastic partial differential equations with nonlinear diffusion coefficients in UMD Banach spaces. [arXiv] (2016), 51 pages.
  • Cox, S., Hutzenthaler, M., Jentzen, A., van Neerven, J., and Welti, T., Convergence in Hölder norms with applications to Monte Carlo methods in infinite dimensions. [arXiv] (2016), 38 pages. To appear in IMA J. Num. Anal.
  • Jacobe de Naurois, L., Jentzen, A., and Welti, T., Weak convergence rates for spatial spectral Galerkin approximations of semilinear stochastic wave equations with multiplicative noise. [arXiv] (2015), 27 pages. Accepted in Appl. Math. Optim.
  • Jentzen, A. and Kurniawan, R., Weak convergence rates for Euler-type approximations of semilinear stochastic evolution equations with nonlinear diffusion coefficients. [arXiv] (2015), 51 pages.
  • Andersson, A., Jentzen, A., and Kurniawan, R., Existence, uniqueness, and regularity for stochastic evolution equations with irregular initial values. [arXiv] (2015), 31 pages. Revision requested from J. Math. Anal. Appl..
  • Hutzenthaler, M., Jentzen, A. and Noll, M., Strong convergence rates and temporal regularity for Cox-Ingersoll-Ross processes and Bessel processes with accessible boundaries. [arXiv] (2014), 32 pages.
  • Hutzenthaler, M. and Jentzen, A., On a perturbation theory and on strong convergence rates for stochastic ordinary and partial differential equations with non-globally monotone coefficients. [arXiv] (2014), 41 pages. To appear in The Annals of Probability.
  • Cox, S., Hutzenthaler, M. and Jentzen, A., Local Lipschitz continuity in the initial value and strong completeness for nonlinear stochastic differential equations. [arXiv] (2013), 54 pages. Revision requested from Memoires of the American Mathematical Society.

Publications

  • Jentzen, A., von Wurstemberger, P., Lower error bounds for the stochastic gradient descent optimization algorithm: Sharp convergence rates for slowly and fast decaying learning rates. Journal of Complexity ? (2019). [arXiv].
  • Beck, C., E, W., Jentzen, A., Machine learning approximation algorithms for high-dimensional fully nonlinear partial differential equations and second-order backward stochastic differential equations. Journal of Nonlinear Science 29 (2019), 1563-1619. [arXiv].
  • Jentzen, A., Lindner, F., Pusnik, P., On the Alekseev-Gröbner formula in Banach spaces. AIMS 24 (2019), 4475-4511. [arXiv].
  • Becker, S., Cheridito, P., Jentzen, A., Deep optimal stopping. JMLR 20 (2019), 1-25. [arXiv].
  • Jentzen, A. and Pusnik, P., Strong convergence rates for an explicit numerical approximation method for stochastic evolution equations with non-globally Lipschitz continuous nonlinearities. IMA Journal of Numerical Analysis ? (2019), 1-38. [arXiv].
  • E, W., Hutzenthaler, M., Jentzen, A., Kruse, T., On multilevel Picard numerical approximations for high-dimensional nonlinear parabolic partial differential equations and high-dimensional nonlinear backward stochastic differential equations. Journal of Scientific Computing 79 (2019), 1534-1571. [arXiv].
  • Jentzen, A., Salimova, D., Welti, T., Strong convergence for explicit space-time discrete numerical approximation methods for stochastic Burgers equations. Journal of Mathematical Analysis and Applications 469 (2019), 661-704. [arXiv].
  • Conus, D., Jentzen, A. and Kurniawan, R., Weak convergence rates of spectral Galerkin approximations for SPDEs with nonlinear diffusion coefficients. Ann. Appl. Probab. 29 (2019), 653-716. [arXiv].
  • Hefter, M., Jentzen, A., On arbitrarily slow convergence rates for strong numerical approximations of Cox-Ingersoll-Ross processes and squared Bessel processes. Finance and Stochastics. 23 (2019), 139-172. [arXiv].
  • Da Prato, G., Jentzen, A. and Röckner, M., A mild Ito formula for SPDEs. Trans. Amer. Math. Soc. ? (2019). [arXiv].
  • Hutzenthaler, M., Jentzen, A., Salimova, D., Strong convergence of full-discrete nonlinearity-truncated accelerated exponential euler-type approximations for stochastic KuramotoÐSivashinsky equations. Comm. Math. Sci. 16 (2018), 1489-1529. [arXiv].
  • Han, J., Jentzen, A., E, W., Solving high-dimensional partial differential equations using deep learning. Proc. Natl. Acad. Sci. 115 (2018), 8505-8510. [arXiv].
  • Jacobe de Naurois, L., Jentzen, A., and Welti, T., Lower bounds for weak approximation errors for spatial spectral Galerkin approximations of stochastic wave equations. Stochastic Partial Differential Equations and Related Fields. 229 (2018), 237-248. [arXiv].
  • Cox, S., Jentzen, A., Kurniawan, R., and Pusnik, P., On the mild Ito formula in Banach spaces. Discrete Contin. Dyn. Syst. Ser. B. 23 (2018), 2217-2243. [arXiv].
  • Andersson, A., Hefter, M., Jentzen, A., and Kurniawan, R., Regularity properties for solutions of infinite dimensional Kolmogorov equations in Hilbert spaces. Potential Analysis ? (2018). [arXiv].
  • Jentzen, A. and Pusnik, P., Exponential moments for numerical approximations of stochastic partial differential equations. SPDE: Anal. and Comp. 6 (2018), 565-617. [arXiv].
  • Becker, S. and Jentzen, A., Strong convergence rates for nonlinearity-truncated Euler-type approximations of stochastic Ginzburg-Landau equations. Stochastic Process. Appl. 129 (2018), 28-69. [arXiv].
  • Hutzenthaler, M., Jentzen, A. and Wang, X., Exponential integrability properties of numerical approximation processes for nonlinear stochastic differential equations. Math. Comp. 87 (2018), 1353-1413. [arXiv].
  • E, W., Han, J., Jentzen, A., Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations. Communications in Mathematics and Statistics 5 (2017), 349-380. [arXiv].
  • Gerencsér, M., Jentzen, A., and Salimova, D., On stochastic differential equations with arbitrarily slow convergence rates for strong approximation in two space dimensions. Proc. Roy. Soc. London A 473 (2017). [arXiv].
  • Andersson, A., Jentzen, A., Kurniawan, R., and Welti, T., On the differentiability of solutions of stochastic evolution equations with respect to their initial values. Nonlinear Analysis 162 (2017), 128-161. [arXiv].
  • Jentzen, A., Müller-Gronbach, T., and Yaroslavtseva, L., On stochastic differential equations with arbitrary slow convergence rates for strong approximation. Commun. Math. Sci. 14 (2016), no. 6, 1477-1500. [arXiv].
  • Becker, S., Jentzen, A. and Kloeden, P. E., An exponential Wagner-Platen type scheme for SPDEs. SIAM J. Numer. Anal. 54 (2016), no. 4, 2389-2426. [arXiv].
  • E, W., Jentzen, A. and Shen, H., Renormalized powers of Ornstein-Uhlenbeck processes and well-posedness of stochastic Ginzburg-Landau equations. Nonlinear Anal. 142 (2016), no. 142, 152-193. [arXiv].
  • Hutzenthaler, M. and Jentzen, A., Numerical approximations of stochastic differential equations with non-globally Lipschitz continuous coefficients. Mem. Amer. Math. Soc. 236 (2015), no. 1112, 99 pages. [arXiv].
  • Jentzen, A. and Röckner, M., A Milstein scheme for SPDEs. Found. Comput. Math. 15 (2015), no. 2, 313-362. [arXiv].
  • Hairer, M., Hutzenthaler, M. and Jentzen, A., Loss of regularity for Kolmogorov equations. Ann. Probab. 43 (2015), no. 2, 468-527. [arXiv].
  • Hutzenthaler, M., Jentzen, A. and Kloeden, P. E., Divergence of the multilevel Monte Carlo Euler method for nonlinear stochastic differential equations. Ann. Appl. Probab. 23 (2013), no. 5, 1913-1966. [arXiv].
  • Blömker, D. and Jentzen, A., Galerkin approximations for the stochastic Burgers equation. SIAM J. Numer. Anal. 51 (2013), no. 1, 694-715. [arXiv].
  • Hutzenthaler, M., Jentzen, A. and Kloeden, P. E., Strong convergence of an explicit numerical method for SDEs with non-globally Lipschitz continuous coefficients. Ann. Appl. Probab. 22 (2012), no. 4, 1611-1641. [arXiv].
  • Jentzen, A. and Röckner, M., Regularity analysis for stochastic partial differential equations with nonlinear multiplicative trace class noise. J. Differential Equations 252 (2012), no. 1, 114-136. [arXiv].
  • Hutzenthaler, M. and Jentzen, A., Convergence of the stochastic Euler scheme for locally Lipschitz coefficients. Found. Comput. Math. 11 (2011), no. 6, 657-706. [arXiv].
  • 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 Winkel, G., Efficient simulation of nonlinear parabolic SPDEs with additive noise. Ann. Appl. Probab. 21 (2011), no. 3, 908-950. [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. Proc. R. Soc. A 467 (2011), no. 2130, 1563-1576. [arXiv].
  • Jentzen, A., Higher order pathwise numerical approximations of SPDEs with additive noise. SIAM J. Numer. Anal. 49 (2011), no. 2, 642-667.
  • Jentzen, A., Taylor expansions of solutions of stochastic partial differential equations. Discrete Contin. Dyn. Syst. Ser. 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. Ann. Probab. 38 (2010), no. 2, 532-569. [arXiv].
  • 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 Trans. Automat. Control 55 (2010), no. 1, 179-184.
  • Jentzen, A. and Kloeden, P. E., A unified existence and uniqueness theorem for stochastic evolution equations. Bull. Aust. Math. Soc. 81 (2010), no. 1, 33-46.
  • Jentzen, A. and Kloeden, P. E., The numerical approximation of stochastic partial differential equations. Milan J. Math. 77 (2009), no. 1, 205-244.
  • 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.
  • Jentzen, A., Pathwise numerical approximations of SPDEs with additive noise under non-global Lipschitz coefficients. Potential Anal. 31 (2009), no. 4, 375-404.
  • Jentzen, A. and Kloeden, P. E., Pathwise Taylor schemes for random ordinary differential equations. BIT 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. Numer. Math. 112 (2009), no. 1, 41-64.
  • Jentzen, A. and Kloeden, P. E., Overcoming the order barrier in the numerical approximation of stochastic partial differential equations with additive space-time noise. Proc. R. Soc. A 465 (2009), no. 2102, 649-667.
  • Jentzen, A. and Neuenkirch, A., A random Euler scheme for Carathéodory differential equations. J. Comput. Appl. Math. 224 (2009), no. 1, 346-359.
  • Kloeden, P. E. and Jentzen, A., Pathwise convergent higher order numerical schemes for random ordinary differential equations. Proc. R. Soc. A 463 (2007), no. 2087, 2929-2944.

Theses

  • Jentzen, A., Taylor Expansions for Stochastic Partial Differential Equations. PhD thesis (2009), Frankfurt University, Germany.
  • Jentzen, A., Numerische Verfahren hoher Ordnung für zufällige Differentialgleichungen. Diploma thesis (2007), Frankfurt University, Germany.