Arnulf Jentzen
ETH Zurich
Address:
Prof. Dr. Arnulf Jentzen
Seminar for Applied Mathematics
Department of Mathematics
ETH Zurich
Rämistrasse 101
8092 Zürich
Switzerland
Office: Room HG G 58.1
Fon (Secretariat): +41 44 633 4766
Fax: +41 44 632 1104
Office hour: on appointment
Email: arnulf.jentzen (at) sam.math.ethz.ch
Homepage: http://www.ajentzen.de
Homepage at ETH Zurich: https://www.math.ethz.ch/sam/theinstitute/people.html?u=jentzena
Born: November 1983 (age 35)
Links:
[Profile on Google Scholar]
[Profile on ResearchGate]
[Profile on MathSciNet]
[ETH Webmail]
[Stochastic Computation Workshop 2017]
Last update of this homepage: May 21st, 2019
Research group
Current members of the research group
 Christian Beck (PhD student at ETH Zurich, DMATH, joint supervision with Prof. Dr. Norbert Hungerbühler)
 Prof. Dr. Arnulf Jentzen (Head of the research group)
 Dr. Ariel Neufeld (Postdoc/Fellow at ETH Zurich, DMATH, joint mentoring with Prof. Dr. Patrick Cheridito)
 Primoz Pusnik (PhD Student at ETH Zurich, DMATH, Seminar for Applied Mathematics)
 Diyora Salimova (PhD student at ETH Zurich, DMATH, Seminar for Applied Mathematics)
 Philippe von Wurstemberger (PhD student at ETH Zurich, DMATH, Seminar for Applied Mathematics)
 Timo Welti (PhD Student at ETH Zurich, DMATH, Seminar for Applied Mathematics)
 Dr. Larisa Yaroslavtseva (Postdoc/Fellow at ETH Zurich, DMATH, Seminar for Applied Mathematics)
Former members of the research group
 Dr. Sebastian Becker (former PhD student, joint supervision with Prof. Dr. Peter E. Kloeden, 20102017, now at Zenai AG, Zurich, Switzerland)
 Prof. Dr. Sonja Cox (former Postdoc/Fellow, 20122014, now tenuretrack Assistant Professor at the University of Amsterdam)
 Dr. Fabian Hornung (former Postdoc/Fellow, 20182018, now Postdoc at the Karlsruhe Institute of Technology)
 Dr. Raphael Kruse
(former Postdoc, 20122014, now Head of the Junior Reseach Group
"Uncertainty Quantification" at the Technical University of Berlin)
 Dr. Ryan Kurniawan (former PhD student, 20142018, now Associate at Market Risk Analytics at Morgan Stanley UK Ltd.)
 Prof. Dr. Michaela Szoelgyenyi (former Postdoc/Fellow, 20172018, now full professor at the University of Klagenfurt)
Research areas
 Machine learning (mathematics for deep learning,
stochastic gradient descent methods, deep artificial neural networks,
empirical risk minimization)
 Stochastic analysis (stochastic calculus, wellposedness and regularity analysis for
stochastic ordinary and partial differential equations)
 Numerical analysis (computational stochastics/stochastic numerics, computational finance)
 Analysis of partial differential equations (wellposedness and regularity analysis for partial differential equations)
Editorial boards affiliations
Preprints and publications that did not yet appear
on MathSciNet
 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., MuellerGronbach, 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 nonconvex 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 linearimplicit 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 ItôAlekseevGröbner formula for stochastic differential equations.
[arXiv] (2018), 29 pages.
 Jentzen, A., Lindner, F., Pusnik, P.,
Exponential moment bounds and strong convergence rates for tamedtruncated numerical approximations of stochastic convolutions.
[arXiv] (2018), 25 pages.
 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.
 Jentzen, A., Lindner, F., Pusnik, P.,
On the AlekseevGröbner formula in Banach spaces.
[arXiv] (2018), 36 pages.
 Elbraechter, D., Grohs, P., Jentzen, A., Schwab, C.,
DNN Expression Rate Analysis of Highdimensional 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 BlackScholes 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 BlackScholes partial
differential equations.
[arXiv] (2018), 124 pages.
 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), 27 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.
 Becker, S., Cheridito, P., Jentzen, A.,
Deep optimal stopping.
[arXiv] (2018), 18 pages.
 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.
[arXiv] (2018), 42 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 spacetime discrete numerical approximations of stochastic AllenCahn equations.
[arXiv] (2017), 104 pages.
 Beck, C., E, W., Jentzen, A.,
Machine learning approximation algorithms for highdimensional fully nonlinear partial differential equations and secondorder backward stochastic differential equations.
[arXiv] (2017), 56 pages. Accepted in J. Nonlinear Sci.
 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 highdimensional 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 Eulertype 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 CoxIngersollRoss 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 nonglobally monotone coefficients.
[arXiv] (2014), 41 pages.
 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 according to MathSciNet
 Jentzen, A. and Pusnik, P.,
Strong convergence rates for an explicit numerical approximation method for stochastic evolution equations with nonglobally Lipschitz continuous nonlinearities.
IMA Journal of Numerical Analysis ? (2019), 138.
[arXiv].
 E, W., Hutzenthaler, M., Jentzen, A., Kruse, T.,
On multilevel Picard numerical approximations for highdimensional nonlinear parabolic partial differential equations and highdimensional nonlinear backward stochastic differential equations.
Journal of Scientific Computing 79 (2019), 15341571.
[arXiv].
 Jentzen, A., Salimova, D., Welti, T.,
Strong convergence for explicit spacetime discrete numerical approximation methods for stochastic Burgers equations.
Journal of Mathematical Analysis and Applications 469 (2019), 661704.
[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), 653716.
[arXiv].
 Hefter, M., Jentzen, A.,
On arbitrarily slow convergence rates for strong numerical approximations of CoxIngersollRoss processes and squared Bessel processes.
Finance and Stochastics. 23 (2019), 139172.
[arXiv].
 Hutzenthaler, M., Jentzen, A., Salimova, D.,
Strong convergence of fulldiscrete nonlinearitytruncated accelerated exponential eulertype approximations for stochastic KuramotoÐSivashinsky equations.
Comm. Math. Sci. 16 (2018), 14891529.
[arXiv].
 Han, J., Jentzen, A., E, W.,
Solving highdimensional partial differential equations using deep learning.
Proc. Natl. Acad. Sci. 115 (2018), 85058510.
[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), 237248.
[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), 22172243.
[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), 565617.
[arXiv].
 Becker, S. and Jentzen, A.,
Strong convergence rates for nonlinearitytruncated Eulertype approximations of stochastic GinzburgLandau equations.
Stochastic Process. Appl. 129 (2018), 2869.
[arXiv].
 Da Prato, G., Jentzen, A. and Röckner, M.,
A mild Ito formula for SPDEs.
Trans. Amer. Math. Soc. ? (2018).
[arXiv].
 Hutzenthaler, M., Jentzen, A. and Wang, X.,
Exponential integrability properties of numerical approximation processes for nonlinear stochastic differential equations.
Math. Comp. 87 (2018), 13531413.
[arXiv].
 E, W., Han, J., Jentzen, A.,
Deep learningbased numerical methods for highdimensional
parabolic partial differential equations and backward stochastic
differential equations.
Communications in Mathematics and Statistics 5 (2017), 349380.
[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), 128161.
[arXiv].
 Jentzen, A., MüllerGronbach, T., and Yaroslavtseva, L.,
On stochastic differential equations with arbitrary slow convergence rates for strong approximation.
Commun. Math. Sci. 14 (2016), no. 6, 14771500.
[arXiv].
 Becker, S., Jentzen, A. and Kloeden, P. E.,
An exponential WagnerPlaten type scheme for SPDEs.
SIAM J. Numer. Anal. 54 (2016), no. 4, 23892426.
[arXiv].
 E, W., Jentzen, A. and Shen, H.,
Renormalized powers of OrnsteinUhlenbeck processes and wellposedness of stochastic GinzburgLandau equations.
Nonlinear Anal.
142 (2016), no. 142, 152193. [arXiv].
 Hutzenthaler, M. and Jentzen, A.,
Numerical approximations of stochastic differential equations with
nonglobally 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, 313362.
[arXiv].
 Hairer, M., Hutzenthaler, M. and Jentzen, A.,
Loss of regularity for Kolmogorov equations.
Ann. Probab.
43 (2015), no. 2, 468527.
[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, 19131966. [arXiv].
 Blömker, D. and Jentzen, A.,
Galerkin approximations for the
stochastic Burgers equation.
SIAM J. Numer. Anal.
51 (2013), no. 1, 694715.
[arXiv].
 Hutzenthaler, M., Jentzen, A. and Kloeden, P. E.,
Strong convergence of an explicit numerical method
for SDEs with nonglobally Lipschitz
continuous coefficients.
Ann. Appl. Probab.
22 (2012), no. 4, 16111641.
[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, 114136.
[arXiv].
 Hutzenthaler, M. and Jentzen, A.,
Convergence of the
stochastic Euler scheme
for locally Lipschitz coefficients.
Found.
Comput. Math.
11 (2011), no. 6, 657706.
[arXiv].

Jentzen, A. and Kloeden, P. E.,
Taylor Approximations for Stochastic
Partial Differential Equations.
CBMSNSF 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, 908950.
[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 nonglobally Lipschitz continuous
coefficients.
Proc. R. Soc. A
467 (2011), no. 2130, 15631576.
[arXiv].
 Jentzen, A.,
Higher order pathwise numerical approximations
of SPDEs with additive noise.
SIAM J. Numer. Anal.
49 (2011),
no. 2, 642667.
 Jentzen, A.,
Taylor expansions of
solutions of stochastic partial
differential equations.
Discrete Contin.
Dyn. Syst. Ser. B
14 (2010), no. 2, 515557.
[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, 532569.
[arXiv].
 Jentzen, A., Leber, F., Schneisgen, D., Berger, A.
and Siegmund., S.,
An improved maximum allowable
transfer interval for Lpstability
of networked control systems.
IEEE Trans. Automat. Control
55 (2010),
no. 1, 179184.
 Jentzen, A. and Kloeden, P. E.,
A unified existence and uniqueness theorem
for stochastic evolution equations.
Bull. Aust. Math. Soc.
81 (2010),
no. 1, 3346.
 Jentzen, A. and Kloeden, P. E.,
The numerical approximation of stochastic partial
differential equations.
Milan
J. Math.
77 (2009), no. 1, 205244.
 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, 140161, London Mathematical Society
Lecture Note Series, 363,
Cambridge University Press, Cambridge, 2009.
 Jentzen, A.,
Pathwise numerical approximations of
SPDEs with additive noise under nonglobal Lipschitz coefficients.
Potential
Anal. 31 (2009), no. 4, 375404.
 Jentzen, A. and Kloeden, P. E.,
Pathwise Taylor schemes
for random ordinary differential
equations.
BIT 49 (2009), no. 1, 113140.
 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, 4164.
 Jentzen, A. and Kloeden, P. E.,
Overcoming the order barrier
in the numerical approximation of
stochastic partial differential
equations with additive
spacetime noise.
Proc. R. Soc. A
465 (2009),
no. 2102, 649667.
 Jentzen, A. and Neuenkirch, A.,
A random Euler scheme for
Carathéodory differential equations.
J.
Comput. Appl. Math.
224 (2009), no. 1, 346359.
 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, 29292944.
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.
