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% Encoding: UTF-8
@Preamble{
"\def\polhk#1{\setbox0=\hbox{#1}{\ooalign{\hidewidth
\lower1.5ex\hbox{`}\hidewidth\crcr\unhbox0}}}"
}
}
@article {MR4265903,
title={{A fast iterative PDE-based algorithm for feedback controls of nonsmooth mean-field control problems}},
author={Reisinger, Christoph and Stockinger, Wolfgang and Zhang, Yufei},
journal={arXiv:2108.06740},
year={2021},
pages={26 pages}
}
}
@article {MR4265654,
AUTHOR = {Gu, Yiqi and Yang, Haizhao and Zhou, Chao},
TITLE = {Select{N}et: self-paced learning for high-dimensional partial
differential equations},
JOURNAL = {J. Comput. Phys.},
FJOURNAL = {Journal of Computational Physics},
VOLUME = {441},
YEAR = {2021},
PAGES = {Paper No. 110444, 18},
ISSN = {0021-9991},
MRCLASS = {65N99 (65M12 65M99 65N12)},
MRNUMBER = {4265654},
DOI = {10.1016/j.jcp.2021.110444},
URL = {https://doi.org/10.1016/j.jcp.2021.110444},
}
@article{nusken2021solving,
title={{Solving high-dimensional Hamilton--Jacobi--Bellman PDEs using neural networks: perspectives from the theory of controlled diffusions and measures on path space}},
author={N{\"u}sken, Nikolas and Richter, Lorenz},
journal={Partial Differ. Equ. Appl.},
volume={2},
number={4},
pages={1--48},
year={2021},
publisher={Springer}
}
@article{kapllani2020deep,
title={Deep Learning algorithms for solving high dimensional nonlinear Backward Stochastic Differential Equations},
author={Kapllani, Lorenc and Teng, Long},
journal={arXiv:2010.01319},
year={2020},
pages={21 pages},
}
@article{chassagneux2021learning,
title={{A learning scheme by sparse grids and Picard approximations for semilinear parabolic PDEs}},
author={Chassagneux, Jean-Fran{\c{c}}ois and Chen, Junchao and Frikha, Noufel and Zhou, Chao},
journal={arXiv:2102.12051},
year={2021},
pages={56 pages},
}
@article{beck2020overview,
title={An overview on deep learning-based approximation methods for partial differential equations},
author={Beck, Christian and Hutzenthaler, Martin and Jentzen, Arnulf and Kuckuck, Benno},
journal={arXiv:2012.12348},
year={2020},
pages={23 pages},
note={\emph{Revisions requested from Discrete Contin. Dyn. Syst. Ser. B}},
}
@article{hutzenthaler2021multilevel,
title={{Multilevel Picard approximations for McKean-Vlasov stochastic differential equations}},
author={Hutzenthaler, Martin and Kruse, Thomas and Nguyen, Tuan Anh},
journal={arXiv:2103.10870},
year={2021},
pages={14 pages},
}
@article{kloeden2007pathwise,
title={The pathwise convergence of approximation schemes for stochastic differential equations},
author={Kloeden, Peter E and Neuenkirch, Andreas},
journal={LMS J.\ Comput.\ Math.},
volume={10},
pages={235--253},
year={2007},
publisher={Cambridge University Press}
}
@article{rio2009moment,
title={Moment inequalities for sums of dependent random variables under projective conditions},
author={Rio, Emmanuel},
journal={J.\ Theor.\ Probab.},
volume={22},
number={1},
pages={146--163},
year={2009},
publisher={Springer}
}
@article{FlandoliLuoRicci2020,
title = "A numerical approach to Kolmogorov equation in high dimension based on Gaussian analysis",
journal = "Journal of Mathematical Analysis and Applications",
pages = "124505",
year = "2020",
issn = "0022-247X",
doi = "https://doi.org/10.1016/j.jmaa.2020.124505",
url = "http://www.sciencedirect.com/science/article/pii/S0022247X20306673",
author = "Franco Flandoli and Dejun Luo and Cristiano Ricci",
keywords = "Kolmogorov equation, Numerical solution, Iteration schema, Gaussian process",
abstract = "For Kolmogorov equations associated to finite dimensional stochastic differential equations (SDEs) in high dimension, a numerical method alternative to Monte Carlo simulations is proposed. The structure of the SDE is inspired by stochastic Partial Differential Equations (SPDE) and thus contains an underlying Gaussian process which is the key of the algorithm. A series development of the solution in terms of iterated integrals of the Gaussian process is given, it is proved to converge - also in the infinite dimensional limit - and it is numerically tested in a number of examples."
}
@Techreport{GononSchwab2020,
author = {L. Gonon and Ch. Schwab},
title = {Deep ReLU network expression rates for option prices in high-dimensional, exponential Lévy models},
institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
number = {2020-52},
address = {Switzerland},
url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2020/2020-52.pdf },
year = {2020}
}
@misc{SalvadorOosterleeVanDerMeer2020,
title={{Financial option valuation by unsupervised learning with artificial neural networks}},
author={Beatriz Salvador and Cornelis W. Oosterlee and Remco van der Meer},
year={2020},
journal={arXiv:2005.12059},
primaryClass={q-fin.CP},
pages={23 pages}
}
@article{Essis-BrettonGaillardetz2020,
title={{Fast Lower and Upper Estimates for the Price of Constrained Multiple Exercise American Options by Single Pass Lookahead Search and Nearest-Neighbor Martingale}},
author={Nicolas Essis-Breton and Patrice Gaillardetz},
year={2020},
journal={arXiv:2002.11258},
primaryClass={q-fin.CP},
pages={64 pages}
}
@article{BeckerCheriditoJentzen2020,
title={Pricing and Hedging American-Style Options with Deep Learning},
volume={13},
ISSN={1911-8074},
url={http://dx.doi.org/10.3390/jrfm13070158},
DOI={10.3390/jrfm13070158},
number={7},
journal={Journal of Risk and Financial Management},
publisher={MDPI AG},
author={Becker, Sebastian and Cheridito, Patrick and Jentzen, Arnulf},
year={2020},
month={Jul},
pages={158}
}
@ARTICLE {HutzenthalerJentzenKruse2019,
AUTHOR = {Hutzenthaler, Martin and Jentzen, Arnulf and Kruse, Thomas and
Nguyen, Tuan Anh},
TITLE = {A proof that rectified deep neural networks overcome the curse
of dimensionality in the numerical approximation of semilinear
heat equations},
JOURNAL = {Partial Differ. Equ. Appl.},
FJOURNAL = {Partial Differential Equations and Applications},
VOLUME = {1},
YEAR = {2020},
NUMBER = {2},
PAGES = {Paper No. 10},
ISSN = {2662-2963},
MRCLASS = {65C99},
MRNUMBER = {4292849},
DOI = {10.1007/s42985-019-0006-9},
URL = {https://doi.org/10.1007/s42985-019-0006-9},
}
}
@article {MR3767958,
AUTHOR = {E, Weinan and Yu, Bing},
TITLE = {The deep {R}itz method: a deep learning-based numerical
algorithm for solving variational problems},
JOURNAL = {Commun. Math. Stat.},
FJOURNAL = {Communications in Mathematics and Statistics},
VOLUME = {6},
YEAR = {2018},
NUMBER = {1},
PAGES = {1--12},
ISSN = {2194-6701},
MRCLASS = {35J20 (35J91 65N30)},
MRNUMBER = {3767958},
DOI = {10.1007/s40304-018-0127-z},
URL = {https://doi.org/10.1007/s40304-018-0127-z},
}
@book{jones2001lebesgue,
title={Lebesgue integration on {E}uclidean space},
author={Jones, Frank},
year={2001},
publisher={Jones \& Bartlett Learning}
}
@article{bercher2020weak,
title={Weak error analysis for stochastic gradient descent optimization algorithms},
author={Bercher, Aritz and Gonon, Lukas and Jentzen, Arnulf and Salimova, Diyora},
journal={arXiv:2007.02723},
year={2020},
pages={123 pages}
}
@article{grohs2018proof,
title={A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of {B}lack-{S}choles partial differential equations},
author={Grohs, Philipp and Hornung, Fabian and Jentzen, Arnulf and Von Wurstemberger, Philippe},
journal={arXiv:1809.02362},
note={\emph{To appear in Mem.\ Amer.\ Math.\ Soc.}},
year={2018},
pages={124 pages},
}
@article{cox2016convergence,
title={{Convergence in H{\"o}lder norms with applications to Monte Carlo methods in infinite dimensions}},
author={Cox, Sonja and Hutzenthaler, Martin and Jentzen, Arnulf and van Neerven, Jan and Welti, Timo},
journal={IMA Journal of Numerical Analysis}
}
@article{CheriditoEtAl2020,
title={Non-convergence of stochastic gradient descent in the training of deep neural networks},
author={Patrick Cheridito and Arnulf Jentzen and Florian Rossmannek},
journal={arXiv:2006.07075},
year={2020},
PAGES = {12 pages},
}
@article{beck2020nonlinear,
title={{On nonlinear Feynman--Kac formulas for viscosity solutions of semilinear parabolic partial differential equations}},
author={Beck, Christian and Hutzenthaler, Martin and Jentzen, Arnulf},
JOURNAL = {Stoch. Dyn.},
FJOURNAL = {Stochastics and Dynamics},
year={2021},
publisher={World Scientific},
note={doi:10.1142/S0219493721500489},
}
@article{SankararamanEtAl2019,
title={{The Impact of Neural Network Overparameterization on Gradient Confusion and Stochastic Gradient Descent}},
author={Karthik A. Sankararaman and Soham De, Zheng Xu and W. Ronny Huang and Tom Goldstein},
journal={arXiv:1904.06963},
year={2019},
PAGES = {37 pages},
}
@article{SchaulZhangLeCun12,
author = {Schaul, Tom and Zhang, Sixin and LeCun, Yann},
title = {No More Pesky Learning Rates},
journal = {Proceedings of the 30th
International Conference on Machine Learning (ICML)},
year = {2013},
volume={28},
number={3},
pages = {343--351},
}
@article{HutzenthalerJentzenKruse201912,
title={{Overcoming the curse of dimensionality in the numerical approximation of parabolic partial differential equations with gradient-dependent nonlinearities}},
author={Martin Hutzenthaler and Arnulf Jentzen and Thomas Kruse},
journal={Found.\ Comput.\ Math.},
year={2021},
pages={1--62}
}
@article{NueskenRichter2020,
title={{Solving high-dimensional Hamilton-Jacobi-Bellman PDEs using neural networks: perspectives from the theory of controlled diffusions and measures on path space}},
author={Nikolas N{\"u}sken and Lorenz Richter},
journal={arXiv:2005.05409},
year={2020},
pages={40 pages}
}
@article {BeckerBraunwarthEtAl2020,
AUTHOR = {Becker, Sebastian and Braunwarth, Ramon and Hutzenthaler,
Martin and Jentzen, Arnulf and von Wurstemberger, Philippe},
TITLE = {Numerical simulations for full history recursive multilevel
{P}icard approximations for systems of high-dimensional
partial differential equations},
JOURNAL = {Commun. Comput. Phys.},
FJOURNAL = {Communications in Computational Physics},
VOLUME = {28},
YEAR = {2020},
NUMBER = {5},
PAGES = {2109--2138},
ISSN = {1815-2406},
MRCLASS = {65M75 (35R60 60H15)},
MRNUMBER = {4188531},
DOI = {10.4208/cicp.oa-2020-0130},
URL = {https://doi.org/10.4208/cicp.oa-2020-0130},
}
@article{beck2020nested,
title={Nonlinear {M}onte {C}arlo methods with polynomial runtime for high-dimensional iterated nested expectations},
author={Beck, Christian and Jentzen, Arnulf and Kruse, Thomas},
journal={arXiv:2009.13989},
year={2020}
}
@article{BeckGononJentzen2020,
title={{Overcoming the curse of dimensionality in the numerical approximation of high-dimensional semilinear elliptic partial differential equations}},
author={Christian Beck and Lukas Gonon and Arnulf Jentzen},
journal={arXiv:2003.00596},
year={2020},
pages={50 pages},
}
@article{JentzenMazzonettoSalimova2018,
title={Existence and uniqueness properties for solutions of a class of {B}anach space valued evolution equations},
author={Jentzen, Arnulf and Mazzonetto, Sara and Salimova, Diyora},
journal={arXiv:1812.06859},
year={2018},
pages={28 pages}
}
@article{MazzonettoSalimova2020,
title={{Existence, uniqueness, and numerical approximations for stochastic Burgers equations}},
author={Sara Mazzonetto and Diyora Salimova},
journal={Stoch. Anal. Appl.},
year={2020},
pages={24 pages}
}
@article{BeckHutzenthalerJentzen2020,
title={On nonlinear Feynman--Kac formulas for viscosity solutions of semilinear parabolic partial differential equations},
author={Beck, Christian and Hutzenthaler, Martin and Jentzen, Arnulf},
journal={Stochastics and Dynamics},
pages={2150048},
year={2021},
publisher={World Scientific}
}
@book {LiuRoeckner2015SPDEs,
AUTHOR = {Liu, Wei and R\"{o}ckner, Michael},
TITLE = {Stochastic partial differential equations: an introduction},
SERIES = {Universitext},
PUBLISHER = {Springer, Cham},
YEAR = {2015},
PAGES = {vi+266},
ISBN = {978-3-319-22353-7; 978-3-319-22354-4},
MRCLASS = {60H15 (35Q30 35Q53 35R60 60-01 60H05 60H10 60J25)},
MRNUMBER = {3410409},
MRREVIEWER = {Mark C. Veraar},
DOI = {10.1007/978-3-319-22354-4},
URL = {https://doi.org/10.1007/978-3-319-22354-4},
}
@article{GononGrohsEtAl2019,
title={{Uniform error estimates for artificial neural network approximations for heat equations}},
author={Lukas Gonon and Philipp Grohs and Arnulf Jentzen and David Kofler and David {\v S}i{\v s}ka},
journal={arXiv:1911.09647},
year={2019},
pages={70 pages}
}
@article {HenryOudjaneEtAl2019,
AUTHOR = {Henry-Labord\`ere, Pierre and Oudjane, Nadia and Tan, Xiaolu and
Touzi, Nizar and Warin, Xavier},
TITLE = {Branching diffusion representation of semilinear {PDE}s and
{M}onte {C}arlo approximation},
JOURNAL = {Ann. Inst. Henri Poincar\'{e} Probab. Stat.},
FJOURNAL = {Annales de l'Institut Henri Poincar\'{e} Probabilit\'{e}s et
Statistiques},
VOLUME = {55},
YEAR = {2019},
NUMBER = {1},
PAGES = {184--210},
ISSN = {0246-0203},
MRCLASS = {60J85 (35K58 35R25 60H07 65C05)},
MRNUMBER = {3901645},
DOI = {10.1214/17-aihp880},
URL = {https://doi.org/10.1214/17-aihp880},
}
@article{GilesJentzenWelti2019,
title={{Generalised multilevel Picard approximations}},
author={Michael B. Giles and Arnulf Jentzen and Timo Welti},
journal={arXiv:1911.03188},
year={2019},
pages={61 pages},
}
@article {DereichMuellerGronbach2019,
AUTHOR = {Dereich, Steffen and M\"{u}ller-Gronbach, Thomas},
TITLE = {General multilevel adaptations for stochastic approximation
algorithms of {R}obbins-{M}onro and {P}olyak-{R}uppert type},
JOURNAL = {Numer. Math.},
FJOURNAL = {Numerische Mathematik},
VOLUME = {142},
YEAR = {2019},
NUMBER = {2},
PAGES = {279--328},
ISSN = {0029-599X},
MRCLASS = {62L20 (60J10 65C05)},
MRNUMBER = {3941932},
DOI = {10.1007/s00211-019-01024-y},
URL = {https://doi.org/10.1007/s00211-019-01024-y},
}
@article {BottouCurtisNocedal2018,
AUTHOR = {Bottou, L\'{e}on and Curtis, Frank E. and Nocedal, Jorge},
TITLE = {Optimization methods for large-scale machine learning},
JOURNAL = {SIAM Rev.},
FJOURNAL = {SIAM Review},
VOLUME = {60},
YEAR = {2018},
NUMBER = {2},
PAGES = {223--311},
ISSN = {0036-1445},
MRCLASS = {65K05 (68Q25 68T05 90C06 90C30 90C90)},
MRNUMBER = {3797719},
MRREVIEWER = {Boualem Alleche},
DOI = {10.1137/16M1080173},
URL = {https://doi.org/10.1137/16M1080173},
}
@article {BassiRebay1997,
AUTHOR = {Bassi, F. and Rebay, S.},
TITLE = {A high-order accurate discontinuous finite element method for
the numerical solution of the compressible {N}avier-{S}tokes
equations},
JOURNAL = {J. Comput. Phys.},
FJOURNAL = {Journal of Computational Physics},
VOLUME = {131},
YEAR = {1997},
NUMBER = {2},
PAGES = {267--279},
ISSN = {0021-9991},
MRCLASS = {76M10 (65M60)},
MRNUMBER = {1433934},
DOI = {10.1006/jcph.1996.5572},
URL = {https://doi.org/10.1006/jcph.1996.5572},
}
@article {Arnold1982,
AUTHOR = {Arnold, Douglas N.},
TITLE = {An interior penalty finite element method with discontinuous
elements},
JOURNAL = {SIAM J. Numer. Anal.},
FJOURNAL = {SIAM Journal on Numerical Analysis},
VOLUME = {19},
YEAR = {1982},
NUMBER = {4},
PAGES = {742--760},
ISSN = {0036-1429},
MRCLASS = {65N30},
MRNUMBER = {664882},
DOI = {10.1137/0719052},
URL = {https://doi.org/10.1137/0719052},
}
@book {GiraultRaviart1986,
AUTHOR = {Girault, Vivette and Raviart, Pierre-Arnaud},
TITLE = {Finite element methods for {N}avier-{S}tokes equations},
SERIES = {Springer Series in Computational Mathematics},
VOLUME = {5},
NOTE = {Theory and algorithms},
PUBLISHER = {Springer-Verlag, Berlin},
YEAR = {1986},
PAGES = {x+374},
ISBN = {3-540-15796-4},
MRCLASS = {65N30 (65-02 76-08)},
MRNUMBER = {851383},
MRREVIEWER = {Max D. Gunzburger},
DOI = {10.1007/978-3-642-61623-5},
URL = {https://doi.org/10.1007/978-3-642-61623-5},
}
@article {Clement1975,
AUTHOR = {Cl\'{e}ment, Ph.},
TITLE = {Approximation by finite element functions using local
regularization},
JOURNAL = {Rev. Fran\c{c}aise Automat. Informat. Recherche Op\'{e}rationnelle
S\'{e}r.},
FJOURNAL = {RAIRO Analyse Num\'{e}rique},
VOLUME = {9},
YEAR = {1975},
NUMBER = {{\rm R}-2},
PAGES = {77--84},
ISSN = {0399-0516},
MRCLASS = {65N15},
MRNUMBER = {0400739},
MRREVIEWER = {Stephen Hilbert},
}
@book {Hughes1987,
AUTHOR = {Hughes, Thomas J. R.},
TITLE = {The finite element method},
NOTE = {Linear static and dynamic finite element analysis,
With the collaboration of Robert M. Ferencz and Arthur M.
Raefsky},
PUBLISHER = {Prentice Hall, Inc., Englewood Cliffs, NJ},
YEAR = {1987},
PAGES = {xxviii+803},
ISBN = {0-13-317025-X},
MRCLASS = {65-01 (73-01 73-08 73K25)},
MRNUMBER = {1008473},
MRREVIEWER = {P. Rochus},
}
@book {BrennerScott1994,
AUTHOR = {Brenner, Susanne C. and Scott, L. Ridgway},
TITLE = {The mathematical theory of finite element methods},
SERIES = {Texts in Applied Mathematics},
VOLUME = {15},
PUBLISHER = {Springer-Verlag, New York},
YEAR = {1994},
PAGES = {xii+294},
ISBN = {0-387-94193-2},
MRCLASS = {65-01 (46N40 65L60 65M60 65N30 73V05 76M10)},
MRNUMBER = {1278258},
MRREVIEWER = {J. T. Oden},
DOI = {10.1007/978-1-4757-4338-8},
URL = {https://doi.org/10.1007/978-1-4757-4338-8},
}
@book {ChorinMarsden1993,
AUTHOR = {Chorin, Alexandre J. and Marsden, Jerrold E.},
TITLE = {A mathematical introduction to fluid mechanics},
SERIES = {Texts in Applied Mathematics},
VOLUME = {4},
EDITION = {Third},
PUBLISHER = {Springer-Verlag, New York},
YEAR = {1993},
PAGES = {xii+169},
ISBN = {0-387-97918-2},
MRCLASS = {76-02 (35Qxx)},
MRNUMBER = {1218879},
DOI = {10.1007/978-1-4612-0883-9},
URL = {https://doi.org/10.1007/978-1-4612-0883-9},
}
@article {BeckHornungEtAl2019,
AUTHOR = {Beck, Christian and Hornung, Fabian and Hutzenthaler, Martin
and Jentzen, Arnulf and Kruse, Thomas},
TITLE = {Overcoming the curse of dimensionality in the numerical
approximation of {A}llen-{C}ahn partial differential equations
via truncated full-history recursive multilevel {P}icard
approximations},
JOURNAL = {J. Numer. Math.},
FJOURNAL = {Journal of Numerical Mathematics},
VOLUME = {28},
YEAR = {2020},
NUMBER = {4},
PAGES = {197--222},
ISSN = {1570-2820},
MRCLASS = {60H30 (65C05 65M75)},
MRNUMBER = {4191733},
DOI = {10.1515/jnma-2019-0074},
URL = {https://doi.org/10.1515/jnma-2019-0074},
}
@article {HutzenthalerPricing2019,
AUTHOR = {Hutzenthaler, Martin and Jentzen, Arnulf and von
Wurstemberger, Philippe},
TITLE = {Overcoming the curse of dimensionality in the approximative
pricing of financial derivatives with default risks},
JOURNAL = {Electron. J. Probab.},
FJOURNAL = {Electronic Journal of Probability},
VOLUME = {25},
YEAR = {2020},
PAGES = {Paper No. 101, 73},
MRCLASS = {60H35 (35K58 60H10 60H30 91G20)},
MRNUMBER = {4144883},
DOI = {10.1007/s13253-019-00378-y},
URL = {https://doi.org/10.1007/s13253-019-00378-y},
}
@article{Wang2018,
title={{An efficient explicit full discrete scheme for strong approximation of stochastic Allen-Cahn equation}},
author={Wang, Xiaojie},
journal={arXiv:1802.09413},
year={2018},
pages={25 pages}
}
@article{ChenGanWang2018arxiv,
title={{Mean-square approximations of L{\'e}vy noise driven SDEs with super-linearly growing diffusion and jump coefficients}},
author= {Chen, Ziheng and Gan, Siqing and Wang, Xiaojie},
journal={arXiv:1812.03069},
year={2018},
pages={19 pages}
}
@article {LiYang2018,
AUTHOR = {Li, Xiaocui and Yang, Xiaoyuan},
TITLE = {Error estimates of finite element methods for fractional
stochastic {N}avier-{S}tokes equations},
JOURNAL = {J. Inequal. Appl.},
FJOURNAL = {Journal of Inequalities and Applications},
YEAR = {2018},
PAGES = {Paper No. 284, 15},
ISSN = {1029-242X},
MRCLASS = {65M60 (26A33 35Q30 35R60 60H15 65M15 65M75 76D06)},
MRNUMBER = {3866691},
DOI = {10.1186/s13660-018-1880-y},
URL = {https://doi.org/10.1186/s13660-018-1880-y},
}
@article {DuanYang2013,
AUTHOR = {Duan, Yuanyuan and Yang, Xiaoyuan},
TITLE = {The finite element method of a {E}uler scheme for stochastic
{N}avier-{S}tokes equations involving the turbulent component},
JOURNAL = {Int. J. Numer. Anal. Model.},
FJOURNAL = {International Journal of Numerical Analysis and Modeling},
VOLUME = {10},
YEAR = {2013},
NUMBER = {3},
PAGES = {727--744},
ISSN = {1705-5105},
MRCLASS = {76D06 (35Q35 35R60 76F55 76M10)},
MRNUMBER = {3064276},
MRREVIEWER = {Azzouz Dermoune},
}
@article {KossiorisZouraris2013,
AUTHOR = {Kossioris, Georgios T. and Zouraris, Georgios E.},
TITLE = {Finite element approximations for a linear
{C}ahn-{H}illiard-{C}ook equation driven by the space
derivative of a space-time white noise},
JOURNAL = {Discrete Contin. Dyn. Syst. Ser. B},
FJOURNAL = {Discrete and Continuous Dynamical Systems. Series B. A Journal
Bridging Mathematics and Sciences},
VOLUME = {18},
YEAR = {2013},
NUMBER = {7},
PAGES = {1845--1872},
ISSN = {1531-3492},
MRCLASS = {65M60 (65C20 65M15 65M75)},
MRNUMBER = {3066325},
MRREVIEWER = {B\"{u}lent Karas\"{o}zen},
DOI = {10.3934/dcdsb.2013.18.1845},
URL = {https://doi.org/10.3934/dcdsb.2013.18.1845},
}
@article {Gazeau2014,
AUTHOR = {Gazeau, Maxime},
TITLE = {Probability and pathwise order of convergence of a
semidiscrete scheme for the stochastic {M}anakov equation},
JOURNAL = {SIAM J. Numer. Anal.},
FJOURNAL = {SIAM Journal on Numerical Analysis},
VOLUME = {52},
YEAR = {2014},
NUMBER = {1},
PAGES = {533--553},
ISSN = {0036-1429},
MRCLASS = {65N06 (35Q55 35R60 60H15 65N75)},
MRNUMBER = {3166967},
MRREVIEWER = {G. B. Loghmani},
DOI = {10.1137/13090924X},
URL = {https://doi.org/10.1137/13090924X},
}
@article {GlattTemamWang2017,
AUTHOR = {Glatt-Holtz, Nathan and Temam, Roger and Wang, Chuntian},
TITLE = {Time discrete approximation of weak solutions to stochastic
equations of geophysical fluid dynamics and applications},
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YEAR = {2018},
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volume={},
number={},
pages={1701-1708},
keywords={face recognition;image representation;neural nets;DeepFace;human-level performance;face verification;face recognition;alignment step;representation step;3D face modeling;piecewise affine transformation;face representation;deep neural network;labeled faces in the wild;LFW dataset;Face;Three-dimensional displays;Training;Face recognition;Agriculture;Solid modeling;Shape},
doi={10.1109/CVPR.2014.220},
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author = {Hu, Baotian and Lu, Zhengdong and Li, Hang and Chen, Qingcai},
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booktitle = {Proceedings of the 27th International Conference on Neural Information Processing Systems - Volume 2},
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pages = {2042--2050},
}
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and Yang, Jie
and Xiao, Jianwei
and Li, Sheng
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title="Face Recognition Based on Deep Learning",
booktitle="Human Centered Computing",
year="2015",
pages="812--820",
abstract="As one of the non-contact biometrics, face representation had been widely used in many circumstances. However conventional methods could no longer satisfy the demand at present, due to its low recognition accuracy and restrictions of many occasions. In this paper, we presented the deep learning method to achieve facial landmark detection and unrestricted face recognition. To solve the face landmark detection problem, this paper proposed a layer-by-layer training method of a deep convolutional neural network to help the convolutional neural network to converge and proposed a sample transformation method to avoid over-fitting. This method had reached an accuracy of 91{\%} on ORL face database. To solve the face recognition problem, this paper proposed a SIAMESE convolutional neural network which was trained on different parts and scales of a face and concatenated the face representation. The face recognition algorithm had reached an accuracy of 91{\%} on ORL and 81{\%} on LFW face database.",
isbn="978-3-319-15554-8"
}
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MRNUMBER = {3039401},
MRREVIEWER = {Edward J. Allen},
DOI = {10.1515/mcma-2013-0001},
URL = {https://doi.org/10.1515/mcma-2013-0001},
}
@article {henry2019branching,
AUTHOR = {Henry-Labord\`ere, Pierre and Oudjane, Nadia and Tan, Xiaolu and
Touzi, Nizar and Warin, Xavier},
TITLE = {Branching diffusion representation of semilinear {PDE}s and
{M}onte {C}arlo approximation},
JOURNAL = {Ann. Inst. Henri Poincar\'{e} Probab. Stat.},
FJOURNAL = {Annales de l'Institut Henri Poincar\'{e} Probabilit\'{e}s et
Statistiques},
VOLUME = {55},
YEAR = {2019},
NUMBER = {1},
PAGES = {184--210},
ISSN = {0246-0203},
MRCLASS = {60J85 (35K58 35R25 60H07 65C05)},
MRNUMBER = {3901645},
DOI = {10.1214/17-aihp880},
URL = {https://doi.org/10.1214/17-aihp880},
}
@article {warin2018nesting,
AUTHOR = {Warin, Xavier},
TITLE = {Nesting {M}onte {C}arlo for high-dimensional non-linear
{PDE}s},
JOURNAL = {Monte Carlo Methods Appl.},
FJOURNAL = {Monte Carlo Methods and Applications},
VOLUME = {24},
YEAR = {2018},
NUMBER = {4},
PAGES = {225--247},
ISSN = {0929-9629},
MRCLASS = {65C05 (35G20 49L25)},
MRNUMBER = {3891748},
MRREVIEWER = {Charles-Edouard Br\'{e}hier},
DOI = {10.1515/mcma-2018-2020},
URL = {https://doi.org/10.1515/mcma-2018-2020},
}
@article{warin2018monte,
title={Monte {C}arlo for high-dimensional degenerated semi linear and full non linear {PDE}s},
author={Warin, Xavier},
journal={arXiv:1805.05078},
year={2018},
pages={23 pages},
}
@article {skorokhod1964branching,
AUTHOR = {Skorohod, A. V.},
TITLE = {Branching diffusion processes},
JOURNAL = {Teor. Verojatnost. i Primenen.},
FJOURNAL = {Akademija Nauk SSSR. Teorija Verojatnoste\u{\i} i ee Primenenija},
VOLUME = {9},
YEAR = {1964},
PAGES = {492--497},
ISSN = {0040-361x},
MRCLASS = {60.67},
MRNUMBER = {0168030},
MRREVIEWER = {K. Dietz},
}
@article {watanabe1965branching,
AUTHOR = {Watanabe, Shinzo},
TITLE = {On the branching process for {B}rownian particles with an
absorbing boundary},
JOURNAL = {J. Math. Kyoto Univ.},
FJOURNAL = {Journal of Mathematics of Kyoto University},
VOLUME = {4},
YEAR = {1965},
PAGES = {385--398},
ISSN = {0023-608X},
MRCLASS = {60.67 (60.62)},
MRNUMBER = {178505},
MRREVIEWER = {P. E. Ney},
DOI = {10.1215/kjm/1250524667},
URL = {https://doi.org/10.1215/kjm/1250524667},
}
@Article{HutzenthalerJentzen2014,
Title = {{On a perturbation theory and on strong convergence rates for stochastic ordinary and partial differential equations with non-globally monotone coefficients}},
Author = {{Hutzenthaler}, M. and {Jentzen}, A.},
Journal = {arXiv:1401.0295},
Year = {2014},
Pages = {41 pages},
Note = {To appear in \textit{The Annals of Probability}},
}
@book {MR3729416,
AUTHOR = {Einsiedler, Manfred and Ward, Thomas},
TITLE = {Functional analysis, spectral theory, and applications},
SERIES = {Graduate Texts in Mathematics},
VOLUME = {276},
PUBLISHER = {Springer, Cham},
YEAR = {2017},
PAGES = {xiv+614},
ISBN = {978-3-319-58539-0; 978-3-319-58540-6},
MRCLASS = {46-01 (35J25 35P10 35P20 37A99 47-01 47A60)},
MRNUMBER = {3729416},
}
@Book{h81,
Title = {{Geometric Theory of Semilinear Parabolic Equations}},
Author = {Daniel Henry},
Publisher = {Springer-Verlag},
Year = {1981},
Note = {348 pages},
Address = {Berlin},
Series = {Lecture Notes in Mathematics},
Volume = {840},
ISBN = {3-540-10557-3},
Mrclass = {35K55 (34G20 58D25)},
Mrnumber = {MR610244 (83j:35084)},
Mrreviewer = {J. A. Goldstein},
Pages = {iv+348}
}
@Book{PrevotRoeckner2007,
AUTHOR = {Claudia Pr{\'e}v{\^o}t and Michael R{\"o}ckner},
TITLE = "A Concise Course on Stochastic Partial Differential Equations",
SERIES = "Lecture Notes in Mathematics",
VOLUME = "1905",
PUBLISHER = "Springer",
ADDRESS = "Berlin",
YEAR = "2007",
PAGES = "vi+144",
ISBN = "978-3-540-70780-6; 3-540-70780-8",
MRCLASS = "60H15 (60-01)",
MRNUMBER = "MR2329435 (2009a:60069)",
MRREVIEWER = "Sandra Cerrai"
}
@Book{Kuehn2004,
Title = {Stochastische Analysis mit Finanzmathematik},
Author = {Christoph K{\"u}hn},
Year = {2004},
Journal = {Lecture notes},
Pages = {86},
Url = {http://www.math.uni-frankfurt.de/~ismi/kuehn/stoch_ana.pdf}
}
@Book{Kallenberg,
Title = {Foundations of modern probability},
Author = {Kallenberg, Olav},
Publisher = {Springer-Verlag},
Year = {2002},
Address = {New York},
Edition = {Second},
Series = {Probability and its Applications (New York)},
ISBN = {0-387-95313-2},
Mrclass = {60-01},
Mrnumber = {1876169 (2002m:60002)},
Mrreviewer = {Klaus D. Schmidt},
Pages = {xx+638}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%From viscosity solutions
@Article{CrandallLions1983,
Title = {Viscosity solutions of {H}amilton-{J}acobi equations},
Author = {Crandall, Michael G. and Lions, Pierre-Louis},
Journal = {Trans. Amer. Math. Soc.},
Year = {1983},
Number = {1},
Pages = {1--42},
Volume = {277},
Coden = {TAMTAM},
Doi = {10.2307/1999343},
Fjournal = {Transactions of the American Mathematical Society},
ISSN = {0002-9947},
Mrclass = {35F20},
Mrnumber = {690039 (85g:35029)},
Mrreviewer = {Moshe Marcus},
Url = {http://dx.doi.org/10.2307/1999343}
}
@book {MR2604525,
AUTHOR = {M\"orters, Peter and Peres, Yuval},
TITLE = {Brownian motion},
SERIES = {Cambridge Series in Statistical and Probabilistic Mathematics},
VOLUME = {30},
NOTE = {With an appendix by Oded Schramm and Wendelin Werner},
PUBLISHER = {Cambridge University Press, Cambridge},
YEAR = {2010},
PAGES = {xii+403},
ISBN = {978-0-521-76018-8},
MRCLASS = {60J65 (28A78 60H05 60J45 60J55 60J67)},
MRNUMBER = {2604525},
MRREVIEWER = {Ren\'e L. Schilling},
DOI = {10.1017/CBO9780511750489},
URL = {https://doi.org/10.1017/CBO9780511750489},
}
@Article{Evans1978,
Title = {A convergence theorem for solutions of nonlinear second-order elliptic equations},
Author = {Evans, Lawrence C.},
Journal = {Indiana Univ. Math. J.},
Year = {1978},
Number = {5},
Pages = {875--887},
Volume = {27},
Coden = {IUMJAB},
Doi = {10.1512/iumj.1978.27.27059},
Fjournal = {Indiana University Mathematics Journal},
ISSN = {0022-2518},
Mrclass = {35J60},
Mrnumber = {503721 (80e:35023)},
Mrreviewer = {P. C. Fife},
Url = {http://dx.doi.org/10.1512/iumj.1978.27.27059}
}
@Article{Evans1980,
Title = {On solving certain nonlinear partial differential equations by accretive operator methods},
Author = {Evans, Lawrence C.},
Journal = {Israel J. Math.},
Year = {1980},
Number = {3-4},
Pages = {225--247},
Volume = {36},
Coden = {ISJMAP},
Doi = {10.1007/BF02762047},
Fjournal = {Israel Journal of Mathematics},
ISSN = {0021-2172},
Mrclass = {35G20 (47H06 47H15)},
Mrnumber = {597451 (82b:35032)},
Mrreviewer = {J. Kolom{\'y}},
Url = {http://dx.doi.org/10.1007/BF02762047}
}
@Article{CrandallLions1981,
Title = {Condition d'unicit\'e pour les solutions g\'en\'eralis\'ees des \'equations de {H}amilton-{J}acobi du premier ordre},
Author = {Crandall, Michael G. and Lions, Pierre-Louis},
Journal = {C. R. Acad. Sci. Paris S\'er. I Math.},
Year = {1981},
Number = {3},
Pages = {183--186},
Volume = {292},
Coden = {CRSMDY},
Fjournal = {Comptes Rendus des S\'eances de l'Acad\'emie des Sciences. S\'erie I. Math\'ematique},
ISSN = {0151-0509},
Mrclass = {49C10},
Mrnumber = {610314 (82c:49020)}
}
@Article{CrandallIshiiLions1992,
Title = {User's guide to viscosity solutions of second order partial differential equations},
Author = {Crandall, Michael G. and Ishii, Hitoshi and Lions, Pierre-Louis},
Journal = {Bull. Amer. Math. Soc. (N.S.)},
Year = {1992},
Number = {1},
Pages = {1--67},
Volume = {27},
Coden = {BAMOAD},
Doi = {10.1090/S0273-0979-1992-00266-5},
Fjournal = {American Mathematical Society. Bulletin. New Series},
ISSN = {0273-0979},
Mrclass = {35J60 (35B05 35D05 35G20)},
Mrnumber = {1118699 (92j:35050)},
Mrreviewer = {P. Szeptycki},
Url = {http://dx.doi.org/10.1090/S0273-0979-1992-00266-5}
}
@Book{Evans2010,
Title = {Partial differential equations},
Author = {Evans, Lawrence C.},
Publisher = {American Mathematical Society},
Year = {2010},
Address = {Providence, RI},
Edition = {Second},
Series = {Graduate Studies in Mathematics},
Volume = {19},
ISBN = {978-0-8218-4974-3},
Mrclass = {35-01},
Mrnumber = {2597943 (2011c:35002)},
Mrreviewer = {Diego M. Maldonado},
Pages = {xxii+749}
}
@article {MR3766391,
AUTHOR = {Hutzenthaler, Martin and Jentzen, Arnulf and Wang, Xiaojie},
TITLE = {Exponential integrability properties of numerical
approximation processes for nonlinear stochastic differential
equations},
JOURNAL = {Math. Comp.},
FJOURNAL = {Mathematics of Computation},
VOLUME = {87},
YEAR = {2018},
NUMBER = {311},
PAGES = {1353--1413},
ISSN = {0025-5718},
MRCLASS = {65C30 (60H35)},
MRNUMBER = {3766391},
DOI = {10.1090/mcom/3146},
URL = {https://doi.org/10.1090/mcom/3146},
}
@Article{Peng2010,
Title = {Nonlinear Expectations and Stochastic Calculus under Uncertainty},
Author = {Shige Peng},
Journal = {arXiv:1002.4546v1},
Year = {2010},
Pages = {149 pages}
}
@article{HairerHutzenthalerJentzenAOP,
author = {Hairer, Martin and Hutzenthaler, Martin and Jentzen, Arnulf},
doi = {10.1214/13-AOP838},
fjournal = {The Annals of Probability},
journal = {Ann. Probab.},
month = {03},
number = {2},
pages = {468--527},
publisher = {The Institute of Mathematical Statistics},
title = {Loss of regularity for Kolmogorov equations},
url = {https://doi.org/10.1214/13-AOP838},
volume = {43},
year = {2015}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Deep learning
@book{GoodfellowBengioCourville16,
title={Deep learning},
author={Goodfellow, Ian and Bengio, Yoshua and Courville, Aaron},
volume={1},
year={2016},
publisher={MIT press Cambridge}
}
@article{LeCunBengioHinton15,
title={Deep learning},
author={LeCun, Yann and Bengio, Yoshua and Hinton, Geoffrey},
journal={nature},
volume={521},
number={7553},
pages={436},
year={2015},
publisher={Nature Publishing Group}
}
@incollection{LeCunBottouOrrMueller98,
title={Efficient backprop},
author={LeCun, Yann and Bottou, L{\'e}on and Orr, Genevieve B and M{\"u}ller, Klaus-Robert},
booktitle={Neural networks: Tricks of the trade},
pages={9--50},
year={1998},
publisher={Springer}
}
% Other articles
@article{DereichGronbach17,
Title = {General multilevel adaptations for stochastic approximation algorithms},
Author = {Steffen Dereich and Thomas Mueller-Gronbach},
Journal = {arXiv:1506.05482},
Year = {2017},
Pages = {33 pages}
}
@article{KingmaBa14,
author = {Diederik P. Kingma and
Jimmy Ba},
title = {Adam: {A} Method for Stochastic Optimization},
journal = {arXiv:1412.6980},
year = {2014},
pages = {15 pages},
url = {http://arxiv.org/abs/1412.6980},
archivePrefix = {arXiv},
eprint = {1412.6980},
timestamp = {Wed, 07 Jun 2017 14:40:52 +0200},
biburl = {http://dblp.org/rec/bib/journals/corr/KingmaB14},
bibsource = {dblp computer science bibliography, http://dblp.org}
}
@article{JentzenWurstemberger2018,
author = {Jentzen, Arnulf and von Wurstemberger, Philippe},
title = {Lower error bounds for the stochastic gradient descent optimization algorithm: Sharp convergence rates for slowly and fast decaying learning rates},
journal = {arXiv:1803.08600},
year = {2018},
pages = {42 pages},
}
@article{Ruder16,
author = {Sebastian Ruder},
title = {An overview of gradient descent optimization algorithms},
journal = {arXiv:1609.04747},
year = {2016},
pages = {12 pages},
timestamp = {Wed, 07 Jun 2017 14:40:06 +0200},
biburl = {http://dblp.uni-trier.de/rec/bib/journals/corr/Ruder16},
bibsource = {dblp computer science bibliography, http://dblp.org}
}
@article{Zeiler12,
author = {Matthew D. Zeiler},
title = {ADADELTA: An Adaptive Learning Rate Method},
journal = {arXiv:1212.5701},
year = {2012},
pages = {6 pages},
url = {http://arxiv.org/abs/1212.5701},
archivePrefix = {arXiv},
eprint = {1212.5701},
timestamp = {Wed, 07 Jun 2017 14:43:02 +0200},
biburl = {http://dblp.org/rec/bib/journals/corr/abs-1212-5701},
bibsource = {dblp computer science bibliography, http://dblp.org}
}
@article{DuchiHazanSinger11,
author = {Duchi, John and Hazan, Elad and Singer, Yoram},
title = {Adaptive Subgradient Methods for Online Learning and Stochastic Optimization},
journal = {J. Mach. Learn. Res.},
issue_date = {2/1/2011},
volume = {12},
month = jul,
year = {2011},
issn = {1532-4435},
pages = {2121--2159},
numpages = {39},
url = {http://dl.acm.org/citation.cfm?id=1953048.2021068},
acmid = {2021068},
publisher = {JMLR.org},
}
@inproceedings{SutskeverMartensDahlHinton2013,
title={On the importance of initialization and momentum in deep learning},
author={Sutskever, Ilya and Martens, James and Dahl, George and Hinton, Geoffrey},
booktitle={International conference on machine learning},
pages={1139--1147},
year={2013}
}
@inproceedings{Nesterov83,
title={A method of solving a convex programming problem with convergence rate O (1/k2)},
author={Nesterov, Yurii},
booktitle={Soviet Mathematics Doklady},
volume={27},
pages={372--376},
year={1983}
}
@article{RobbinsMonro51,
ISSN = {00034851},
URL = {http://www.jstor.org/stable/2236626},
author = {Herbert Robbins and Sutton Monro},
journal = {The Annals of Mathematical Statistics},
number = {3},
pages = {400-407},
publisher = {Institute of Mathematical Statistics},
title = {A Stochastic Approximation Method},
volume = {22},
year = {1951}
}
@article{Polyak64,
title={Some methods of speeding up the convergence of iteration methods},
author={Polyak, Boris T},
journal={USSR Computational Mathematics and Mathematical Physics},
volume={4},
number={5},
pages={1--17},
year={1964},
publisher={Elsevier}
}
@misc{HintonSlides,
title = {Lecture 6e: Rmsprop: Divide the gradient by a running average of its recent magnitude},
author = {Hinton, Geoffrey and Srivastava, Nitish and Swersky, Kevin},
howpublished = {\url{http://www.cs.toronto.edu/~tijmen/csc321/slides/lecture_slides_lec6.pdf}},
note = {26-31, Accessed: 01.12.2017},
pages = {26-31}
}
@book{BoydVandenberghe04 ,
title={Convex optimization},
author={Boyd, Stephen and Vandenberghe, Lieven},
year={2004},
publisher={Cambridge University Press}
}
@book{Nesterov13,
title={Introductory lectures on convex optimization: A basic course},
author={Nesterov, Yurii},
volume={87},
year={2013},
publisher={Springer Science \& Business Media}
}
@article{Bubeck15,
url = {http://dx.doi.org/10.1561/2200000050},
year = {2015},
volume = {8},
journal = {Foundations and Trends{\textregistered} in Machine Learning},
title = {Convex Optimization: Algorithms and Complexity},
doi = {10.1561/2200000050},
issn = {1935-8237},
number = {3-4},
pages = {231-357},
author = {Sébastien Bubeck}
}
@misc{Dozat16,
title={Incorporating nesterov momentum into adam},
author={Dozat, Timothy},
year={2016},
howpublished = "\url{https://openreview.net/forum?id=OM0jvwB8jIp57ZJjtNEZ}",
note = "[Accessed 6-December-2017]"
}
@misc{Dozat15,
title={Incorporating nesterov momentum into adam},
author={Dozat, Timothy},
year={2016},
howpublished = "\url{http://cs229.stanford.edu/proj2015/054_report.pdf}",
note = "[Accessed 6-December-2017]"
}
@article{Qian99,
title={On the momentum term in gradient descent learning algorithms},
author={Qian, Ning},
journal={Neural networks},
volume={12},
number={1},
pages={145--151},
year={1999},
publisher={Elsevier}
}
@article{EHanJentzen17,
AUTHOR = {E, Weinan and Han, Jiequn and Jentzen, Arnulf},
TITLE = {Deep learning-based numerical methods for high-dimensional
parabolic partial differential equations and backward
stochastic differential equations},
JOURNAL = {Commun. Math. Stat.},
FJOURNAL = {Communications in Mathematics and Statistics},
VOLUME = {5},
YEAR = {2017},
NUMBER = {4},
PAGES = {349--380},
ISSN = {2194-6701},
MRCLASS = {65M75 (60H35 65C30)},
MRNUMBER = {3736669},
MRREVIEWER = {Samuel Bernard},
DOI = {10.1007/s40304-017-0117-6},
URL = {https://doi.org/10.1007/s40304-017-0117-6},
}
@article{Silvester2000,
title={Determinants of block matrices},
author={Silvester, John R},
journal={The Mathematical Gazette},
volume={84},
number={501},
pages={460--467},
year={2000},
publisher={JSTOR}
}
@inproceedings{Tropp01,
title={An Elementary Proof of the Spectral Radius Formula for Matrices},
author={Joel A. Tropp},
year ={2001},
note = {\url{http://users.cms.caltech.edu/~jtropp/notes/Tro01-Spectral-Radius.pdf}. [Accessed 16-February-2018]},
}
@article{EinsiedlerWard15,
title={Functional Analysis, Spectral Theory, and Applications},
author={Einsiedler, Manfred and Ward, Thomas},
journal={draft version},
year={2015},
publisher={Springer}
}
@article{Mypaper,
author = {Jentzen, Arnulf and Kuckuck, Benno and Neufeld, Ariel and von Wurstemberger,Philippe},
title = {Strong error analysis for stochastic gradient descent optimization algorithms},
journal = {arXiv:1801.09324},
year = {2018},
pages = {75 pages},
}
%Averaging
@techreport{Ruppert88,
title={Efficient estimations from a slowly convergent Robbins-Monro process},
author={Ruppert, David},
year={1988},
institution={Cornell University Operations Research and Industrial Engineering}
}
@article{PolyakJuditsky92,
title={Acceleration of stochastic approximation by averaging},
author={Polyak, Boris T and Juditsky, Anatoli B},
journal={SIAM Journal on Control and Optimization},
volume={30},
number={4},
pages={838--855},
year={1992},
publisher={SIAM}
}
@article{Polyak90,
title={New stochastic approximation type procedures},
author={Polyak, Boris T},
journal={Automat. i Telemekh},
volume={7},
number={98-107},
pages={2},
year={1990}
}
%websites
@misc{TensorflowMomentum,
title={Tensorflow: Momentum Optimizer},
howpublished = "\url{https://www.tensorflow.org/api_docs/python/tf/train/MomentumOptimizer}",
note = "[Accessed 6-December-2017]"
}
@misc{TensorflowRMSprop,
title={Tensorflow: RMSProp},
howpublished = "\url{https://www.tensorflow.org/api_docs/python/tf/train/RMSPropOptimizer}",
note = "[Accessed 6-December-2017]"
}
@misc{TensorflowNN,
title={Tensorflow: Neural Networks},
howpublished = "\url{https://www.tensorflow.org/api_guides/python/nn}",
note = "[Accessed 06-April-2018]"
}
@misc{TensorflowRelu,
title={Tensorflow: Relu},
howpublished = "\url{https://www.tensorflow.org/api_docs/python/tf/nn/relu}",
note = "[Accessed 06-April-2018]"
}
@misc{TensorflowSigmoid,
title={Tensorflow: Sigmoid},
howpublished = "\url{https://www.tensorflow.org/api_docs/python/tf/sigmoid}",
note = "[Accessed 06-April-2018]"
}
@misc{TensorflowSoftplus,
title={Tensorflow: Softplus},
howpublished = "\url{https://www.tensorflow.org/api_docs/python/tf/nn/softplus}",
note = "[Accessed 06-April-2018]"
}
@misc{TensorflowSoftmax,
title={Tensorflow: Softmax},
howpublished = "\url{https://www.tensorflow.org/api_docs/python/tf/nn/softmax}",
note = "[Accessed 06-April-2018]"
}
@misc{CaffeOptimizer,
title={Caffe: Solver},
howpublished = "\url{http://caffe.berkeleyvision.org/tutorial/solver.html}",
note = "[Accessed 6-December-2017]"
}
@misc{LasagneOptimizer,
title={Lasagne: Updates},
howpublished = "\url{http://lasagne.readthedocs.io/en/latest/modules/updates.html#lasagne.updates.rmsprop}",
note = "[Accessed 6-December-2017]"
}
@misc{NgCoursera,
title={COURSERA: Improving Deep Neural Networks: Hyperparameter tuning, Regularization and Optimization},
author={Ng, Andrew},
howpublished = "\url{https://www.coursera.org/learn/deep-neural-network}",
note = "[Accessed 6-December-2017]"
}
@misc{HintonCoursera,
title={COURSERA: Neural Networks for Machine learning},
author={Hinton, Geoffrey},
howpublished = "\url{https://www.coursera.org/learn/neural-networks}",
note = "[Accessed 6-December-2017]"
}
@misc{WikipediaGronwall,
title={Gronwall's inequality - Wikipedia},
howpublished = "\url{https://en.wikipedia.org/wiki/Gronwall%27s_inequality}",
note = "[Accessed 10-January-2018]"
}
@misc{WikipediaSpectralRadius,
title={Spectral Radius - Wikipedia},
howpublished = "\url{https://en.wikipedia.org/wiki/Spectral_radius#Gelfand's_formula}",
note = "[Accessed 16-February-2018]"
}
@misc{WikipediaAdditionstheorem,
title={List of trigonometric identities - Wikipedia},
howpublished = "\url{https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Angle_sum_and_difference_identities}",
note = "[Accessed 06-March-2018]"
}
@misc{WikipediaLBlog,
title={Lower Bound of Natural Logarithm - Pr$\infty$fWiki},
howpublished = "\url{https://proofwiki.org/wiki/Lower_Bound_of_Natural_Logarithm}",
note = "[Accessed 08-March-2018]"
}
@misc{WikipediaRelu,
title={Rectifier (neural networks) - Wikipedia},
howpublished = "\url{https://en.wikipedia.org/wiki/Rectifier_(neural_networks)}",
note = "[Accessed 06-April-2018]"
}
@misc{WikipediaSoftmax,
title={Softmax function - Wikipedia},
howpublished = "\url{https://en.wikipedia.org/wiki/Softmax_function}",
note = "[Accessed 06-April-2018]"
}
@misc{WikipediaSigmoid,
title={Sigmoid function - Wikipedia},
howpublished = "\url{https://en.wikipedia.org/wiki/Sigmoid_function}",
note = "[Accessed 06-April-2018]"
}
%Reference books
@book{Coleman12,
title={Calculus on normed vector spaces},
author={Coleman, Rodney},
year={2012},
publisher={Springer Science \& Business Media}
}
@book{Petersen12,
title={Linear Algebra},
author={Petersen, P.},
isbn={9781461436119},
lccn={2012938237},
series={Undergraduate Texts in Mathematics},
url={https://books.google.ch/books?id=3klgLwEACAAJ},
year={2012},
publisher={Springer New York}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5
@article{Dockhorn2019,
title={A Discussion on Solving Partial Differential Equations using Neural Networks},
author={Tim Dockhorn},
journal={arXiv:1904.07200},
year={2019},
PAGES = {9 pages},
}
@article {YangZhang2017,
AUTHOR = {Yang, Li and Zhang, Yanzhi},
TITLE = {Convergence of the spectral {G}alerkin method for the
stochastic reaction-diffusion-advection equation},
JOURNAL = {J. Math. Anal. Appl.},
FJOURNAL = {Journal of Mathematical Analysis and Applications},
VOLUME = {446},
YEAR = {2017},
NUMBER = {2},
PAGES = {1230--1254},
ISSN = {0022-247X},
MRCLASS = {65M60 (60H35 65M12 65M75)},
MRNUMBER = {3563032},
MRREVIEWER = {Josep Vives},
DOI = {10.1016/j.jmaa.2016.09.028},
URL = {http://dx.doi.org/10.1016/j.jmaa.2016.09.028},
}
@article {KamraniBloemker2017,
AUTHOR = {Kamrani, Minoo and Bl\"omker, Dirk},
TITLE = {Pathwise convergence of a numerical method for stochastic
partial differential equations with correlated noise and local
{L}ipschitz condition},
JOURNAL = {J. Comput. Appl. Math.},
FJOURNAL = {Journal of Computational and Applied Mathematics},
VOLUME = {323},
YEAR = {2017},
PAGES = {123--135},
ISSN = {0377-0427},
MRCLASS = {60H35 (60H10 60H15 65M12 65M60 65M75)},
MRNUMBER = {3649320},
DOI = {10.1016/j.cam.2017.04.012},
URL = {http://dx.doi.org/10.1016/j.cam.2017.04.012},
}
@article {Kamrani2015,
AUTHOR = {Kamrani, Minoo},
TITLE = {Numerical solution of stochastic fractional differential
equations},
JOURNAL = {Numer. Algorithms},
FJOURNAL = {Numerical Algorithms},
VOLUME = {68},
YEAR = {2015},
NUMBER = {1},
PAGES = {81--93},
ISSN = {1017-1398},
MRCLASS = {65C30 (26A33 60H15)},
MRNUMBER = {3296700},
MRREVIEWER = {Gabriela Mircea},
DOI = {10.1007/s11075-014-9839-7},
URL = {http://dx.doi.org/10.1007/s11075-014-9839-7},
}
@article {HairerMatetski2016,
AUTHOR = {Hairer, M. and Matetski, K.},
TITLE = {Optimal rate of convergence for stochastic {B}urgers-type
equations},
JOURNAL = {Stoch. Partial Differ. Equ. Anal. Comput.},
FJOURNAL = {Stochastic Partial Differential Equations. Analysis and
Computations},
VOLUME = {4},
YEAR = {2016},
NUMBER = {2},
PAGES = {402--437},
ISSN = {2194-0401},
MRCLASS = {35R60 (35Q53 60H15 60H99)},
MRNUMBER = {3498987},
DOI = {10.1007/s40072-015-0067-5},
URL = {http://dx.doi.org/10.1007/s40072-015-0067-5},
}
@article {FurihataKovacsLarssonLindgren2018,
AUTHOR = {Furihata, Daisuke and Kov\'acs, Mih\'aly and Larsson, Stig and
Lindgren, Fredrik},
TITLE = {Strong {C}onvergence of a {F}ully {D}iscrete {F}inite
{E}lement {A}pproximation of the {S}tochastic
{C}ahn--{H}illiard {E}quation},
JOURNAL = {SIAM J. Numer. Anal.},
FJOURNAL = {SIAM Journal on Numerical Analysis},
VOLUME = {56},
YEAR = {2018},
NUMBER = {2},
PAGES = {708--731},
ISSN = {0036-1429},
MRCLASS = {60H15 (60H35 65M75)},
MRNUMBER = {3780120},
DOI = {10.1137/17M1121627},
URL = {https://doi.org/10.1137/17M1121627},
}
@article {CuiHongLiuZhou2019,
AUTHOR = {Cui, Jianbo and Hong, Jialin and Liu, Zhihui and Zhou, Weien},
TITLE = {Strong convergence rate of splitting schemes for stochastic
nonlinear {S}chr\"{o}dinger equations},
JOURNAL = {J. Differential Equations},
FJOURNAL = {Journal of Differential Equations},
VOLUME = {266},
YEAR = {2019},
NUMBER = {9},
PAGES = {5625--5663},
ISSN = {0022-0396},
MRCLASS = {60H35 (35Q55 35R60 60G05 60H15)},
MRNUMBER = {3912762},
DOI = {10.1016/j.jde.2018.10.034},
URL = {https://doi.org/10.1016/j.jde.2018.10.034},
}
@article {CuiHongLiu2017,
AUTHOR = {Cui, Jianbo and Hong, Jialin and Liu, Zhihui},
TITLE = {Strong convergence rate of finite difference approximations
for stochastic cubic {S}chr\"odinger equations},
JOURNAL = {J. Differential Equations},
FJOURNAL = {Journal of Differential Equations},
VOLUME = {263},
YEAR = {2017},
NUMBER = {7},
PAGES = {3687--3713},
ISSN = {0022-0396},
MRCLASS = {60H35 (35Q55 35R60 60H15 65M06 65M12)},
MRNUMBER = {3670034},
DOI = {10.1016/j.jde.2017.05.002},
URL = {http://dx.doi.org/10.1016/j.jde.2017.05.002},
}
@article{AnderssonJentzen2017,
title={Existence, uniqueness, and regularity for stochastic evolution equations with irregular initial values},
author={Adam Andersson and Arnulf Jentzen and Ryan Kurniawan},
journal={arXiv:1512.06899},
year={2015},
PAGES = {35 pages},
note = {Revision requested from \emph{J.\ Math.\ Anal.\ Appl.}}
}
@article {AnderssonKruse2017,
AUTHOR = {Andersson, Adam and Kruse, Raphael},
TITLE = {Mean-square convergence of the {BDF}2-{M}aruyama and backward
{E}uler schemes for {SDE} satisfying a global monotonicity
condition},
JOURNAL = {BIT},
FJOURNAL = {BIT. Numerical Mathematics},
VOLUME = {57},
YEAR = {2017},
NUMBER = {1},
PAGES = {21--53},
ISSN = {0006-3835},
MRCLASS = {65C30 (60H35)},
MRNUMBER = {3608309},
MRREVIEWER = {Carlos M. Mora Gonz\'{a}lez},
DOI = {10.1007/s10543-016-0624-y},
URL = {https://doi.org/10.1007/s10543-016-0624-y},
}
@article {HefterJentzen2019,
AUTHOR = {Hefter, Mario and Jentzen, Arnulf},
TITLE = {On arbitrarily slow convergence rates for strong numerical
approximations of {C}ox-{I}ngersoll-{R}oss processes and
squared {B}essel processes},
JOURNAL = {Finance Stoch.},
FJOURNAL = {Finance and Stochastics},
VOLUME = {23},
YEAR = {2019},
NUMBER = {1},
PAGES = {139--172},
ISSN = {0949-2984},
MRCLASS = {60H10 (65C30 91G60)},
MRNUMBER = {3898403},
DOI = {10.1007/s00780-018-0375-5},
URL = {https://doi.org/10.1007/s00780-018-0375-5},
}
@book {Novak2010,
AUTHOR = {Novak, Erich and Wo\'{z}niakowski, Henryk},
TITLE = {Tractability of multivariate problems. {V}olume {II}:
{S}tandard information for functionals},
SERIES = {EMS Tracts in Mathematics},
VOLUME = {12},
PUBLISHER = {European Mathematical Society (EMS), Z\"{u}rich},
YEAR = {2010},
PAGES = {xviii+657},
ISBN = {978-3-03719-084-5},
MRCLASS = {46N40 (41A55 41A63 47A50 47N40 65Y20 68Q25)},
MRNUMBER = {2676032},
MRREVIEWER = {Arthur G. Werschulz},
DOI = {10.4171/084},
URL = {https://doi.org/10.4171/084},
}
@book {Bellman1957,
AUTHOR = {Bellman, Richard},
TITLE = {Dynamic programming},
PUBLISHER = {Princeton University Press, Princeton, N. J.},
YEAR = {1957},
PAGES = {xxv+342},
MRCLASS = {90.0X},
MRNUMBER = {0090477},
MRREVIEWER = {J. Kiefer},
}
@article{HutzenthalerKruse2017,
title={Multilevel {P}icard Approximations of High-Dimensional Semilinear Parabolic Differential Equations with Gradient-Dependent Nonlinearities},
author={Hutzenthaler, Martin and Kruse, Thomas},
journal={SIAM J.\ Numer.\ Anal.},
volume={58},
number={2},
pages={929--961},
year={2020},
publisher={SIAM}
}
@article{HutzenthalerJentzenKruse2018,
author = {Hutzenthaler, Martin and Jentzen, Arnulf and Kruse, Thomas and Nguyen, Tuan Anh and von Wurstemberger, Philippe },
title = {Overcoming the curse of dimensionality in the numerical approximation of semilinear parabolic partial differential equations},
fjournal = {Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences},
journal = {Proc. Roy. Soc. London A},
volume = {476},
number = {2244},
pages = {20190630},
year = {2020},
doi = {10.1098/rspa.2019.0630},
URL = {https://royalsocietypublishing.org/doi/abs/10.1098/rspa.2019.0630},
eprint = {https://royalsocietypublishing.org/doi/pdf/10.1098/rspa.2019.0630}
,
abstract = { For a long time it has been well-known that high-dimensional linear parabolic partial differential equations (PDEs) can be approximated by Monte Carlo methods with a computational effort which grows polynomially both in the dimension and in the reciprocal of the prescribed accuracy. In other words, linear PDEs do not suffer from the curse of dimensionality. For general semilinear PDEs with Lipschitz coefficients, however, it remained an open question whether these suffer from the curse of dimensionality. In this paper we partially solve this open problem. More precisely, we prove in the case of semilinear heat equations with gradient-independent and globally Lipschitz continuous nonlinearities that the computational effort of a variant of the recently introduced multilevel Picard approximations grows at most polynomially both in the dimension and in the reciprocal of the required accuracy. }
}
@article {EHutzenthaler2019,
AUTHOR = {E, Weinan and Hutzenthaler, Martin and Jentzen, Arnulf and
Kruse, Thomas},
TITLE = {On multilevel {P}icard numerical approximations for
high-dimensional nonlinear parabolic partial differential
equations and high-dimensional nonlinear backward stochastic
differential equations},
JOURNAL = {J. Sci. Comput.},
FJOURNAL = {Journal of Scientific Computing},
VOLUME = {79},
YEAR = {2019},
NUMBER = {3},
PAGES = {1534--1571},
ISSN = {0885-7474},
MRCLASS = {65C05 (65C30 65M99 91G60)},
MRNUMBER = {3946468},
DOI = {10.1007/s10915-018-00903-0},
URL = {https://doi.org/10.1007/s10915-018-00903-0},
}
@article{EHutzenthaler2016,
title={{Multilevel Picard iterations for solving smooth semilinear parabolic heat equations}},
author={Weinan E and Martin Hutzenthaler and Arnulf Jentzen and Thomas Kruse},
journal={arXiv:1607.03295},
note={\emph{Accepted in Partial Differ. Equ. Appl.}},
year={2016},
pages={19 pages},
}
@article {Kruse2014,
AUTHOR = {Kruse, Raphael},
TITLE = {Optimal error estimates of {G}alerkin finite element methods
for stochastic partial differential equations with
multiplicative noise},
JOURNAL = {IMA J. Numer. Anal.},
FJOURNAL = {IMA Journal of Numerical Analysis},
VOLUME = {34},
YEAR = {2014},
NUMBER = {1},
PAGES = {217--251},
ISSN = {0272-4979},
MRCLASS = {65M60 (35R60 60H15 65M75)},
MRNUMBER = {3168284},
MRREVIEWER = {Mohammad Asadzadeh},
DOI = {10.1093/imanum/drs055},
URL = {https://doi.org/10.1093/imanum/drs055},
}
@article {Deya2012,
AUTHOR = {Deya, Aur\'{e}lien},
TITLE = {Numerical schemes for rough parabolic equations},
JOURNAL = {Appl. Math. Optim.},
FJOURNAL = {Applied Mathematics and Optimization},
VOLUME = {65},
YEAR = {2012},
NUMBER = {2},
PAGES = {253--292},
ISSN = {0095-4616},
MRCLASS = {65M75 (35R60 60G17 60G22 60H15)},
MRNUMBER = {2891223},
DOI = {10.1007/s00245-011-9157-6},
URL = {https://doi.org/10.1007/s00245-011-9157-6},
}
@article {WiegmannBube2000,
AUTHOR = {Wiegmann, Andreas and Bube, Kenneth P.},
TITLE = {The explicit-jump immersed interface method: finite difference
methods for {PDE}s with piecewise smooth solutions},
JOURNAL = {SIAM J. Numer. Anal.},
FJOURNAL = {SIAM Journal on Numerical Analysis},
VOLUME = {37},
YEAR = {2000},
NUMBER = {3},
PAGES = {827--862},
ISSN = {0036-1429},
MRCLASS = {65N06 (31A10 31C20)},
MRNUMBER = {1740384},
MRREVIEWER = {Zheng Hui Xie},
DOI = {10.1137/S0036142997328664},
URL = {https://doi.org/10.1137/S0036142997328664},
}
@article {Furihata2001,
AUTHOR = {Furihata, Daisuke},
TITLE = {A stable and conservative finite difference scheme for the
{C}ahn-{H}illiard equation},
JOURNAL = {Numer. Math.},
FJOURNAL = {Numerische Mathematik},
VOLUME = {87},
YEAR = {2001},
NUMBER = {4},
PAGES = {675--699},
ISSN = {0029-599X},
MRCLASS = {65M06 (35K55 65M12 82C26)},
MRNUMBER = {1815731},
MRREVIEWER = {Bruno Scheurer},
DOI = {10.1007/PL00005429},
URL = {https://doi.org/10.1007/PL00005429},
}
@article {YusteAcedo2005,
AUTHOR = {Yuste, S. B. and Acedo, L.},
TITLE = {An explicit finite difference method and a new von
{N}eumann-type stability analysis for fractional diffusion
equations},
JOURNAL = {SIAM J. Numer. Anal.},
FJOURNAL = {SIAM Journal on Numerical Analysis},
VOLUME = {42},
YEAR = {2005},
NUMBER = {5},
PAGES = {1862--1874},
ISSN = {0036-1429},
MRCLASS = {45K05 (65M06)},
MRNUMBER = {2139227},
MRREVIEWER = {\v{Z}ivorad Tomovski},
DOI = {10.1137/030602666},
URL = {https://doi.org/10.1137/030602666},
}
@article {Meerschaert2006,
AUTHOR = {Meerschaert, Mark M. and Tadjeran, Charles},
TITLE = {Finite difference approximations for two-sided
space-fractional partial differential equations},
JOURNAL = {Appl. Numer. Math.},
FJOURNAL = {Applied Numerical Mathematics. An IMACS Journal},
VOLUME = {56},
YEAR = {2006},
NUMBER = {1},
PAGES = {80--90},
ISSN = {0168-9274},
MRCLASS = {65M06},
MRNUMBER = {2186432},
MRREVIEWER = {Matthew Edward Hubbard},
DOI = {10.1016/j.apnum.2005.02.008},
URL = {https://doi.org/10.1016/j.apnum.2005.02.008},
}
@article {Babuska1978,
AUTHOR = {Babu\v{s}ka, I. and Rheinboldt, W. C.},
TITLE = {Error estimates for adaptive finite element computations},
JOURNAL = {SIAM J. Numer. Anal.},
FJOURNAL = {SIAM Journal on Numerical Analysis},
VOLUME = {15},
YEAR = {1978},
NUMBER = {4},
PAGES = {736--754},
ISSN = {0036-1429},
MRCLASS = {65J05 (65N30)},
MRNUMBER = {0483395},
MRREVIEWER = {William W. Hager},
DOI = {10.1137/0715049},
URL = {https://doi.org/10.1137/0715049},
}
@book {Braess2007,
AUTHOR = {Braess, Dietrich},
TITLE = {Finite elements},
EDITION = {Third},
NOTE = {Theory, fast solvers, and applications in elasticity theory,
Translated from the German by Larry L. Schumaker},
PUBLISHER = {Cambridge University Press, Cambridge},
YEAR = {2007},
PAGES = {xviii+365},
ISBN = {978-0-521-70518-9; 0-521-70518-5},
MRCLASS = {65N30 (65-02 74S05)},
MRNUMBER = {2322235},
DOI = {10.1017/CBO9780511618635},
URL = {https://doi.org/10.1017/CBO9780511618635},
}
@book {BensoussanLions1982,
AUTHOR = {Bensoussan, Alain and Lions, Jacques-Louis},
TITLE = {Applications of variational inequalities in stochastic
control},
SERIES = {Studies in Mathematics and its Applications},
VOLUME = {12},
PUBLISHER = {North-Holland Publishing Co., Amsterdam-New York},
YEAR = {1982},
PAGES = {xi+564},
ISBN = {0-444-86358-3},
MRCLASS = {49A29 (49-02)},
MRNUMBER = {653144},
}
@inproceedings{uchiyama1993solving,
title={Solving inverse problems in nonlinear {PDE}s by recurrent neural networks},
author={Uchiyama, Tadasu and Sonehara, Noboru},
booktitle={IEEE International Conference on Neural Networks},
pages={99--102},
year={1993},
organization={IEEE}
}
@article {MeadeFernandez1994,
AUTHOR = {Meade, Jr., A. J. and Fern\'{a}ndez, A. A.},
TITLE = {The numerical solution of linear ordinary differential
equations by feedforward neural networks},
JOURNAL = {Math. Comput. Modelling},
FJOURNAL = {Mathematical and Computer Modelling},
VOLUME = {19},
YEAR = {1994},
NUMBER = {12},
PAGES = {1--25},
ISSN = {0895-7177},
MRCLASS = {65L05},
MRNUMBER = {1284175},
DOI = {10.1016/0895-7177(94)90095-7},
URL = {https://doi.org/10.1016/0895-7177(94)90095-7},
}
@article{Lagaris1998ArtificialNN,
title={Artificial neural networks for solving ordinary and partial differential equations},
author={Isaac E. Lagaris and Aristidis Likas and Dimitrios I. Fotiadis},
journal={IEEE transactions on neural networks},
year={1998},
volume={9 (5)},
pages={
987--1000
}
}
@article{GrohsJentzenSalimova2019,
title={{Deep neural network approximations for Monte Carlo algorithms}},
author={Grohs, Philipp and Jentzen, Arnulf and Salimova, Diyora},
journal={arXiv:1908.10828},
year={2019},
PAGES = {45 pages},
}
@article {JentzenSalimovaWeltiBurgers,
AUTHOR = {Jentzen, Arnulf and Salimova, Diyora and Welti, Timo},
TITLE = {Strong convergence for explicit space-time discrete numerical
approximation methods for stochastic {B}urgers equations},
JOURNAL = {J. Math. Anal. Appl.},
FJOURNAL = {Journal of Mathematical Analysis and Applications},
VOLUME = {469},
YEAR = {2019},
NUMBER = {2},
PAGES = {661--704},
ISSN = {0022-247X},
MRCLASS = {60H35 (35Q53 35R60 65M75)},
MRNUMBER = {3860443},
MRREVIEWER = {Romeo I. Negrea},
DOI = {10.1016/j.jmaa.2018.09.032},
URL = {https://doi.org/10.1016/j.jmaa.2018.09.032},
}
@article{JentzenSalimovaWeltiBurgersArxiv,
title={Strong convergence for explicit space-time discrete numerical
approximation methods for stochastic {B}urgers equations},
author={Jentzen, Arnulf and Salimova, Diyora and Welti, Timo},
journal={arXiv:1710.07123},
year={2017},
PAGES = {60 pages},
}
@article {HutzenthalerJentzenSalimova2018,
AUTHOR = {Hutzenthaler, Martin and Jentzen, Arnulf and Salimova, Diyora},
TITLE = {Strong convergence of full-discrete nonlinearity-truncated
accelerated exponential {E}uler-type approximations for
stochastic {K}uramoto-{S}ivashinsky equations},
JOURNAL = {Commun. Math. Sci.},
FJOURNAL = {Communications in Mathematical Sciences},
VOLUME = {16},
YEAR = {2018},
NUMBER = {6},
PAGES = {1489--1529},
ISSN = {1539-6746},
MRCLASS = {60H35 (35Q53 35R60)},
MRNUMBER = {3919602},
DOI = {10.4310/CMS.2018.v16.n6.a2},
URL = {https://doi.org/10.4310/CMS.2018.v16.n6.a2},
}
@incollection {GjerdeHolden1995,
AUTHOR = {Gjerde, Jon and Holden, Helge and \O ksendal, Bernt and Ub\o e,
Jan and Zhang, Tu Sheng},
TITLE = {An equation modelling transport of a substance in a stochastic
medium},
BOOKTITLE = {Seminar on {S}tochastic {A}nalysis, {R}andom {F}ields and
{A}pplications ({A}scona, 1993)},
SERIES = {Progr. Probab.},
VOLUME = {36},
PAGES = {123--134},
PUBLISHER = {Birkh\"{a}user, Basel},
YEAR = {1995},
MRCLASS = {60H15 (35K10 35R60 76R50)},
MRNUMBER = {1360271},
MRREVIEWER = {Krystyna Twardowska},
}
@article {Shandarin1989,
AUTHOR = {Shandarin, S. F. and Zeldovich, Ya. B.},
TITLE = {The large-scale structure of the universe: turbulence,
intermittency, structures in a self-gravitating medium},
JOURNAL = {Rev. Modern Phys.},
FJOURNAL = {Reviews of Modern Physics},
VOLUME = {61},
YEAR = {1989},
NUMBER = {2},
PAGES = {185--220},
ISSN = {0034-6861},
MRCLASS = {85A05 (58C27 58F40)},
MRNUMBER = {989562},
DOI = {10.1103/RevModPhys.61.185},
URL = {https://doi.org/10.1103/RevModPhys.61.185},
}
@incollection {LeGland1992,
AUTHOR = {Le Gland, Fran\c{c}ois},
TITLE = {Splitting-up approximation for {SPDE}s and {SDE}s with
application to nonlinear filtering},
BOOKTITLE = {Stochastic partial differential equations and their
applications ({C}harlotte, {NC}, 1991)},
SERIES = {Lect. Notes Control Inf. Sci.},
VOLUME = {176},
PAGES = {177--187},
PUBLISHER = {Springer, Berlin},
YEAR = {1992},
MRCLASS = {60H15 (60G35 93E11)},
MRNUMBER = {1176783},
DOI = {10.1007/BFb0007332},
URL = {https://doi.org/10.1007/BFb0007332},
}
@book {Wu1996,
AUTHOR = {Wu, Jianhong},
TITLE = {Theory and applications of partial functional-differential
equations},
SERIES = {Applied Mathematical Sciences},
VOLUME = {119},
PUBLISHER = {Springer-Verlag, New York},
YEAR = {1996},
PAGES = {x+429},
ISBN = {0-387-94771-X},
MRCLASS = {35R10 (35Bxx 35K99 35Q80)},
MRNUMBER = {1415838},
MRREVIEWER = {Yang Kuang},
DOI = {10.1007/978-1-4612-4050-1},
URL = {https://doi.org/10.1007/978-1-4612-4050-1},
}
@article {BusenbergMahaffy1985,
AUTHOR = {Busenberg, Stavros and Mahaffy, Joseph},
TITLE = {Interaction of spatial diffusion and delays in models of
genetic control by repression},
JOURNAL = {J. Math. Biol.},
FJOURNAL = {Journal of Mathematical Biology},
VOLUME = {22},
YEAR = {1985},
NUMBER = {3},
PAGES = {313--333},
ISSN = {0303-6812},
MRCLASS = {92A09},
MRNUMBER = {813402},
DOI = {10.1007/BF00276489},
URL = {https://doi.org/10.1007/BF00276489},
}
@Inbook{Levin1978,
author="Levin, Simon A.",
title="Pattern Formation in Ecological Communities",
bookTitle="Spatial Pattern in Plankton Communities",
year="1978",
publisher="Springer US",
address="Boston, MA",
pages="433--465",
abstract="The primary concern of this volume is the study of spatial pattern in plankton communities, but the basic issues are fundamental ones in all ecosystems (Levin, 1976b). The plankton has emerged as perhaps the prototypical system for the discussion of spatial pattern principally because of the ubiquitous nature of patchiness, independent of the scale of observation (Platt and Denman, 1975). Population densities are sufficiently high that pattern is easily detected, and planktonic systems have been intensively studied from a variety of aspects.",
isbn="978-1-4899-2195-6",
doi="10.1007/978-1-4899-2195-6_16",
url="https://doi.org/10.1007/978-1-4899-2195-6_16"
}
@article {MourratWeber2017,
AUTHOR = {Mourrat, Jean-Christophe and Weber, Hendrik},
TITLE = {Convergence of the two-dimensional dynamic {I}sing-{K}ac model
to {$\Phi^4_2$}},
JOURNAL = {Comm. Pure Appl. Math.},
FJOURNAL = {Communications on Pure and Applied Mathematics},
VOLUME = {70},
YEAR = {2017},
NUMBER = {4},
PAGES = {717--812},
ISSN = {0010-3640},
MRCLASS = {82C20 (35R60 60H15 60K35 82C22)},
MRNUMBER = {3628883},
MRREVIEWER = {Arvind Ayyer},
DOI = {10.1002/cpa.21655},
URL = {https://doi.org/10.1002/cpa.21655},
}
@article {Kushner1964,
AUTHOR = {Kushner, Harold J.},
TITLE = {On the differential equations satisfied by conditional
probablitity densities of {M}arkov processes, with
applications},
JOURNAL = {J. Soc. Indust. Appl. Math. Ser. A Control},
FJOURNAL = {Journal of the Society for Industrial and Applied Mathematics.
Series A. Control},
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YEAR = {1964},
PAGES = {106--119},
MRCLASS = {60.75},
MRNUMBER = {0180407},
MRREVIEWER = {A. V. Balakrishnan},
}
@article {Zakai1969,
AUTHOR = {Zakai, Moshe},
TITLE = {On the optimal filtering of diffusion processes},
JOURNAL = {Z. Wahrscheinlichkeitstheorie und Verw. Gebiete},
VOLUME = {11},
YEAR = {1969},
PAGES = {230--243},
MRCLASS = {93.60 (60.00)},
MRNUMBER = {0242552},
MRREVIEWER = {R. S. Bucy},
DOI = {10.1007/BF00536382},
URL = {https://doi.org/10.1007/BF00536382},
}
@book {LeimkuhlerMatthews2015,
AUTHOR = {Leimkuhler, Ben and Matthews, Charles},
TITLE = {Molecular dynamics},
SERIES = {Interdisciplinary Applied Mathematics},
VOLUME = {39},
NOTE = {With deterministic and stochastic numerical methods},
PUBLISHER = {Springer, Cham},
YEAR = {2015},
PAGES = {xxii+443},
ISBN = {978-3-319-16374-1; 978-3-319-16375-8},
MRCLASS = {82Bxx (35Q82 65-02 70-02 82Cxx)},
MRNUMBER = {3362507},
}
@article {HarmsStefanovits2018,
AUTHOR = {Harms, Philipp and Stefanovits, David and Teichmann, Josef and
W\"{u}thrich, Mario V.},
TITLE = {Consistent recalibration of yield curve models},
JOURNAL = {Math. Finance},
FJOURNAL = {Mathematical Finance. An International Journal of Mathematics,
Statistics and Financial Economics},
VOLUME = {28},
YEAR = {2018},
NUMBER = {3},
PAGES = {757--799},
ISSN = {0960-1627},
MRCLASS = {91G30},
MRNUMBER = {3818710},
DOI = {10.1111/mafi.12159},
URL = {https://doi.org/10.1111/mafi.12159},
}
@article {FilipovicTappe2010,
AUTHOR = {Filipovi\'{c}, Damir and Tappe, Stefan and Teichmann, Josef},
TITLE = {Term structure models driven by {W}iener processes and
{P}oisson measures: existence and positivity},
JOURNAL = {SIAM J. Financial Math.},
FJOURNAL = {SIAM Journal on Financial Mathematics},
VOLUME = {1},
YEAR = {2010},
NUMBER = {1},
PAGES = {523--554},
ISSN = {1945-497X},
MRCLASS = {91G30 (60H10 60H15 60H30)},
MRNUMBER = {2669403},
MRREVIEWER = {Guohe Deng},
DOI = {10.1137/090758593},
URL = {https://doi.org/10.1137/090758593},
}
@article{ReisingerZhang2019,
title={Rectified deep neural networks overcome the curse of dimensionality for nonsmooth value functions in zero-sum games of nonlinear stiff systems},
author={Christoph Reisinger and Yufei Zhang},
journal={arXiv:1903.06652},
year={2019},
PAGES = {34 pages},
}
@article{KutyniokPeterseb2019,
title={A Theoretical Analysis of Deep Neural Networks and Parametric {PDE}s},
author={Gitta Kutyniok and Philipp Petersen and Mones Raslan and Reinhold Schneider},
journal={arXiv:1904.00377},
year={2019},
PAGES = {43 pages},
}
@article{BernerGrohsJentzen2018,
title={{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}},
author={Julius Berner and Philipp Grohs and Arnulf Jentzen},
journal={Minor revision requested from SIAM J. Math. Data Sci., arXiv:1809.03062},
year={2018},
PAGES = {35 pages},
}
@article{Raissi2018DeepHP,
title={Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations},
author={Maziar Raissi},
journal={J. Mach. Learn. Res.},
year={2018},
volume={19},
pages={25:1-25:24}
}
@article{PhamWarin2019,
title={{Neural networks-based backward scheme for fully nonlinear PDEs}},
author={Huy{\^e}n Pham and Xavier Warin},
journal={arXiv:1908.00412},
year={2019},
PAGES = {15 pages},
}
@inproceedings{Magill2018NeuralNT,
title={Neural Networks Trained to Solve Differential Equations Learn General Representations},
author={Martin Magill and Faisal Qureshi and Hendrick W. de Haan},
booktitle={Advances in Neural Information Processing Systems},
year={2018},
pages = {4071--4081},
}
@article{LyeMishraRay2019,
title={Deep learning observables in computational fluid dynamics},
author={Lye, Kjetil O. and Siddhartha Mishra and Deep Ray},
journal={arXiv:1903.03040},
year={2019},
PAGES = {57 pages},
}
@InProceedings{LongLuMaDong2018,
title = {{PDE-Net: Learning PDEs from Data}},
author = {Long, Zichao and Lu, Yiping and Ma, Xianzhong and Dong, Bin},
booktitle = {Proceedings of the 35th International Conference on Machine Learning},
pages = {3208--3216},
year = {2018},
pdf = {http://proceedings.mlr.press/v80/long18a/long18a.pdf},
url = {http://proceedings.mlr.press/v80/long18a.html},
abstract = {Partial differential equations (PDEs) play a prominent role in many disciplines of science and engineering. PDEs are commonly derived based on empirical observations. However, with the rapid development of sensors, computational power, and data storage in the past decade, huge quantities of data can be easily collected and efficiently stored. Such vast quantity of data offers new opportunities for data-driven discovery of physical laws. Inspired by the latest development of neural network designs in deep learning, we propose a new feed-forward deep network, called PDE-Net, to fulfill two objectives at the same time: to accurately predict dynamics of complex systems and to uncover the underlying hidden PDE models. Comparing with existing approaches, our approach has the most flexibility by learning both differential operators and the nonlinear response function of the underlying PDE model. A special feature of the proposed PDE-Net is that all filters are properly constrained, which enables us to easily identify the governing PDE models while still maintaining the expressive and predictive power of the network. These constrains are carefully designed by fully exploiting the relation between the orders of differential operators and the orders of sum rules of filters (an important concept originated from wavelet theory). Numerical experiments show that the PDE-Net has the potential to uncover the hidden PDE of the observed dynamics, and predict the dynamical behavior for a relatively long time, even in a noisy environment.}
}
@article{JacquierOumgari2019,
title={{Deep PPDEs for rough local stochastic volatility}},
author={Antoine Jacquier and Mugad Oumgari},
journal={arXiv:1906.02551},
year={2019},
PAGES = {21 pages},
}
@article{HurePhamWarin2019,
title={{Some machine learning schemes for high-dimensional nonlinear PDEs}},
author={C{\^o}me Hur{\'e} and Huy{\^e}n Pham and Xavier Warin},
journal={arXiv:1902.01599},
year={2019},
PAGES = {33 pages},
}
@article{HanLong2018,
title={{Convergence of the Deep BSDE Method for Coupled FBSDEs}},
author={Jiequn Han and Jihao Long},
journal={arXiv:1811.01165},
year={2018},
PAGES = {26 pages},
}
@article{GoudenegeMolent2019,
title={{Machine Learning for Pricing American Options in High Dimension}},
author={Gouden{\`e}ge, Ludovic and Molent, Andrea and Zanette, Antonino},
journal={arXiv:1903.11275},
year={2019},
PAGES = {11 pages},
}
@Article{FujiiTakahashi2019,
author="Fujii, Masaaki
and Takahashi, Akihiko
and Takahashi, Masayuki",
title="Asymptotic Expansion as Prior Knowledge in Deep Learning Method for High dimensional {BSDE}s",
journal="Asia-Pacific Financial Markets",
year="2019",
month="Mar",
day="18",
abstract="We demonstrate that the use of asymptotic expansion as prior knowledge in the ``deep BSDE solver'', which is a deep learning method for high dimensional BSDEs proposed by Weinan et al. (Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations, 2017b. arXiv:1706.04702), drastically reduces the loss function and accelerates the speed of convergence. We illustrate the technique and its implications by using Bergman's model with different lending and borrowing rates as a typical model for FVA as well as a class of solvable BSDEs with quadratic growth drivers. We also present an extension of the deep BSDE solver for reflected BSDEs representing American option prices.",
issn="1573-6946",
doi="10.1007/s10690-019-09271-7",
url="https://doi.org/10.1007/s10690-019-09271-7"
}
@article{Farahmand2017DeepRL,
title={Deep reinforcement learning for partial differential equation control},
author={Farahmand, Amir-massoud and Saleh Nabi and Daniel Nikovski},
journal={2017 American Control Conference (ACC)},
year={2017},
pages={3120-3127}
}
@article {ChanMikaelWarin2019,
AUTHOR = {Chan-Wai-Nam, Quentin and Mikael, Joseph and Warin, Xavier},
TITLE = {Machine learning for semi linear {PDE}s},
JOURNAL = {J. Sci. Comput.},
FJOURNAL = {Journal of Scientific Computing},
VOLUME = {79},
YEAR = {2019},
NUMBER = {3},
PAGES = {1667--1712},
ISSN = {0885-7474},
MRCLASS = {68T05 (65C05 65M99)},
MRNUMBER = {3946472},
DOI = {10.1007/s10915-019-00908-3},
URL = {https://doi.org/10.1007/s10915-019-00908-3},
}
@article{Berg2018AUD,
title={A unified deep artificial neural network approach to partial differential equations in complex geometries},
author={Jens Berg and Kaj Nystr{\"o}m},
journal={Neurocomputing},
year={2018},
volume={317},
pages={28-41}
}
@article{BeckerCheriditoJentzen2019,
title={Solving high-dimensional optimal stopping problems using deep learning},
author={Becker, Sebastian and Cheridito, Patrick and Jentzen, Arnulf and Welti, Timo},
journal={Minor revision requested from European J. Appl. Math., arXiv:1908.01602},
year={2019},
PAGES = {42 pages},
}
@article{BeckBeckerCheridito2019,
title={{Deep splitting method for parabolic PDEs}},
author={Beck, Christian and Becker, Sebastian and Cheridito, Patrick and Jentzen, Arnulf and Neufeld, Ariel},
journal={Revision requested from SIAM J. Sci. Comput., arXiv:1907.03452},
year={2019},
PAGES = {40 pages},
}
@article {Sirignano2018,
AUTHOR = {Sirignano, Justin and Spiliopoulos, Konstantinos},
TITLE = {D{GM: A} deep learning algorithm for solving partial
differential equations},
JOURNAL = {J. Comput. Phys.},
FJOURNAL = {Journal of Computational Physics},
VOLUME = {375},
YEAR = {2018},
PAGES = {1339--1364},
ISSN = {0021-9991},
MRCLASS = {65M99 (68T05)},
MRNUMBER = {3874585},
DOI = {10.1016/j.jcp.2018.08.029},
URL = {https://doi.org/10.1016/j.jcp.2018.08.029},
}
@article {sirignano2017dgm,
AUTHOR = {Sirignano, Justin and Spiliopoulos, Konstantinos},
TITLE = {{D{GM}: A deep learning algorithm for solving partial
differential equations}},
JOURNAL = {J. Comput. Phys.},
FJOURNAL = {Journal of Computational Physics},
VOLUME = {375},
YEAR = {2018},
PAGES = {1339--1364},
ISSN = {0021-9991},
MRCLASS = {65M99 (68T05)},
MRNUMBER = {3874585},
DOI = {10.1016/j.jcp.2018.08.029},
URL = {https://doi.org/10.1016/j.jcp.2018.08.029},
}
@article{mishra2018machine,
title={A machine learning framework for data driven acceleration of computations of differential equations},
author={Mishra, Siddhartha},
journal={arXiv:1807.09519},
year={2018},
PAGES = {23 pages},
}
@article{khoo2017solving,
title={Solving parametric {PDE} problems with artificial neural networks},
author={Khoo, Yuehaw and Lu, Jianfeng and Ying, Lexing},
journal={arXiv:1707.03351},
year={2017},
PAGES = {17 pages},
}
@article {KhooLuYing2019,
AUTHOR = {Khoo, Yuehaw and Lu, Jianfeng and Ying, Lexing},
TITLE = {Solving for high-dimensional committor functions using
artificial neural networks},
JOURNAL = {Res. Math. Sci.},
FJOURNAL = {Research in the Mathematical Sciences},
VOLUME = {6},
YEAR = {2019},
NUMBER = {1},
PAGES = {Paper No. 1, 13},
ISSN = {2522-0144},
MRCLASS = {60H10 (35Q84 35R60 65C30)},
MRNUMBER = {3887217},
DOI = {10.1007/s40687-018-0160-2},
URL = {https://doi.org/10.1007/s40687-018-0160-2},
}
@article{henry2017deep,
title={Deep {Primal}-{Dual} {Algorithm} for {BSDEs}: {Applications} of {Machine} {Learning} to {CVA} and {IM}},
author={Henry-Labord\`ere, Pierre},
year={2017},
pages= {(November 15, 2017), 16 pages},
note= {Available at SSRN: \url{https://ssrn.com/abstract=3071506}},
}
@article {MR3881695,
AUTHOR = {Raissi, M. and Perdikaris, P. and Karniadakis, G. E.},
TITLE = {{Physics-informed neural networks: A deep learning framework
for solving forward and inverse problems involving nonlinear
partial differential equations}},
JOURNAL = {J. Comput. Phys.},
FJOURNAL = {Journal of Computational Physics},
VOLUME = {378},
YEAR = {2019},
PAGES = {686--707},
ISSN = {0021-9991},
MRCLASS = {65M70 (68T05)},
MRNUMBER = {3881695},
DOI = {10.1016/j.jcp.2018.10.045},
URL = {https://doi.org/10.1016/j.jcp.2018.10.045},
}
@article {Han2018PNAS,
AUTHOR = {Han, Jiequn and Jentzen, Arnulf and E, Weinan},
TITLE = {Solving high-dimensional partial differential equations using
deep learning},
JOURNAL = {Proc. Natl. Acad. Sci. USA},
FJOURNAL = {Proceedings of the National Academy of Sciences of the United
States of America},
VOLUME = {115},
YEAR = {2018},
NUMBER = {34},
PAGES = {8505--8510},
ISSN = {0027-8424},
MRCLASS = {35K55 (68T05 91A26 92C20)},
MRNUMBER = {3847747},
DOI = {10.1073/pnas.1718942115},
URL = {https://doi.org/10.1073/pnas.1718942115},
}
@article{fujii2017asymptotic,
title={Asymptotic {Expansion} as {Prior} {Knowledge} in {Deep} {Learning} {Method} for high dimensional {BSDEs}},
author={Fujii, Masaaki and Takahashi, Akihiko and Takahashi, Masayuki},
journal={arXiv:1710.07030},
year={2017},
PAGES = {16 pages},
}
@article {weinan2018deep,
AUTHOR = {E, Weinan and Yu, Bing},
TITLE = {The deep {R}itz method: {A} deep learning-based numerical
algorithm for solving variational problems},
JOURNAL = {Commun. Math. Stat.},
FJOURNAL = {Communications in Mathematics and Statistics},
VOLUME = {6},
YEAR = {2018},
NUMBER = {1},
PAGES = {1--12},
ISSN = {2194-6701},
MRCLASS = {35J20 (35J91 65N30)},
MRNUMBER = {3767958},
DOI = {10.1007/s40304-018-0127-z},
URL = {https://doi.org/10.1007/s40304-018-0127-z},
}
@article {weinan2017deep,
AUTHOR = {E, Weinan and Han, Jiequn and Jentzen, Arnulf},
TITLE = {Deep learning-based numerical methods for high-dimensional
parabolic partial differential equations and backward
stochastic differential equations},
JOURNAL = {Commun. Math. Stat.},
FJOURNAL = {Communications in Mathematics and Statistics},
VOLUME = {5},
YEAR = {2017},
NUMBER = {4},
PAGES = {349--380},
ISSN = {2194-6701},
MRCLASS = {65M75 (60H35 65C30)},
MRNUMBER = {3736669},
MRREVIEWER = {Samuel Bernard},
DOI = {10.1007/s40304-017-0117-6},
URL = {https://doi.org/10.1007/s40304-017-0117-6},
}
@article{BeckerCheridito2019,
author = {Sebastian Becker and Patrick Cheridito and Arnulf Jentzen},
title = {Deep Optimal Stopping},
fjournal = {Journal of Machine Learning Research},
journal={J. Mach. Learn. Res.},
year = {2019},
volume = {20},
number = {74},
pages = {1--25},
url = {http://jmlr.org/papers/v20/18-232.html}
}
@article{Kolmogorov,
AUTHOR = {Beck, Christian and Becker, Sebastian and Grohs, Philipp and
Jaafari, Nor and Jentzen, Arnulf},
TITLE = {Solving the {K}olmogorov {PDE} by means of deep learning},
JOURNAL = {J. Sci. Comput.},
FJOURNAL = {Journal of Scientific Computing},
VOLUME = {88},
YEAR = {2021},
NUMBER = {3},
PAGES = {Paper No. 73, 28},
ISSN = {0885-7474},
MRCLASS = {65C30 (35R60 60H10 62M45 65M75)},
MRNUMBER = {4293960},
DOI = {10.1007/s10915-021-01590-0},
URL = {https://doi.org/10.1007/s10915-021-01590-0},
}
@inproceedings{Zhai2016,
author = {Zhai, Shuangfei and Chang, Keng-hao and Zhang, Ruofei and Zhang, Zhongfei Mark},
title = {DeepIntent: Learning Attentions for Online Advertising with Recurrent Neural Networks},
booktitle = {Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining},
year = {2016},
isbn = {978-1-4503-4232-2},
pages = {1295--1304},
url = {http://doi.acm.org/10.1145/2939672.2939759},
doi = {10.1145/2939672.2939759},
keywords = {BM25, RNN, attention mechanism, deep learning, online advertising, query rewriting, sponsored search},
}
@inproceedings{Wang2017Ad,
author = {Wang, Ruoxi and Fu, Bin and Fu, Gang and Wang, Mingliang},
title = {Deep \& Cross Network for Ad Click Predictions},
booktitle = {Proceedings of the ADKDD'17},
year = {2017},
keywords = {CTR Prediction, Deep Learning, Feature Crossing, Neural Networks},
}
@INPROCEEDINGS{Roy2018,
author={A. Roy and J. Sun and R. Mahoney and L. Alonzi and S. Adams and P. Beling},
booktitle={2018 Systems and Information Engineering Design Symposium (SIEDS)},
title={Deep learning detecting fraud in credit card transactions},
year={2018},
volume={},
number={},
pages={129-134},
keywords={cloud computing;credit transactions;data mining;fraud;learning (artificial intelligence);neural nets;regression analysis;security of data;North American financial institutions;fraudulent activity;credit card fraud detection;real-time transaction details;prevailing fraud detection methods;common fraud detection problems;digital payment systems;deep learning;artificial neural network;credit card transactions;distributed cloud computing environment;Machine learning;Credit cards;Topology;Artificial neural networks;Computer architecture;Network topology;Hidden Markov models;Deep Learning;Fraud Detection;Neural Networks;Sensitivity Analysis},}
@article{CHOUIEKH2018,
title = "ConvNets for Fraud Detection analysis",
journal = "Procedia Computer Science",
volume = "127",
pages = "133--138",
year = "2018",
author = "Alae Chouiekh and EL Hassane Ibn EL Haj",
keywords = "Fraud, Machine learning, Deep learning, CNN"
}
@inproceedings{wu2016stimulated,
author={Chunyang Wu and Penny Karanasou and Mark J.F. Gales and Khe Chai Sim},
title={Stimulated Deep Neural Network for Speech Recognition},
year=2016,
booktitle={Interspeech 2016},
doi={10.21437/Interspeech.2016-580},
url={http://dx.doi.org/10.21437/Interspeech.2016-580},
pages={400--404}
}
@article{JentzenSalimovaWelti2018,
title={A proof that deep artificial neural networks overcome the curse of dimensionality in the numerical approximation of {K}olmogorov partial differential equations with constant diffusion and nonlinear drift coefficients},
author={Jentzen, Arnulf and Salimova, Diyora and Welti, Timo},
journal={arXiv:1809.07321},
year={2018},
pages={48 pages}
}
@Article{jp15,
Title = {{Strong convergence rates for an explicit numerical approximation method for stochastic evolution equations with non-globally Lipschitz continuous nonlinearities}},
Author = {{Jentzen}, A. and {Pu{\v s}nik}, P.},
Journal={arXiv:1504.03523},
Year = {2015},
Pages = {38 pages},
Note = {To appear in \emph{IMA J.\ Num.\ Anal.}},
}
@article{beckergess2017,
title={Strong convergence rates for explicit space-time discrete numerical approximations of stochastic {A}llen-{C}ahn equations},
author={Becker, Sebastian and Gess, Benjamin and Jentzen, Arnulf and Kloeden, Peter E},
journal={arXiv:1711.02423},
year={2017}
}
@article {Yaroslavtseva2017,
AUTHOR = {Yaroslavtseva, Larisa},
TITLE = {On non-polynomial lower error bounds for adaptive strong
approximation of {SDE}s},
JOURNAL = {J. Complexity},
FJOURNAL = {Journal of Complexity},
VOLUME = {42},
YEAR = {2017},
PAGES = {1--18},
ISSN = {0885-064X},
MRCLASS = {65C30 (65Y20)},
MRNUMBER = {3683255},
MRREVIEWER = {Judith M. Ford},
DOI = {10.1016/j.jco.2017.04.002},
URL = {https://doi.org/10.1016/j.jco.2017.04.002},
}
@article {GerencserJentzen2017,
AUTHOR = {Gerencs\'{e}r, M\'{a}t\'{e} and Jentzen, Arnulf and Salimova, Diyora},
TITLE = {On stochastic differential equations with arbitrarily slow
convergence rates for strong approximation in two space
dimensions},
JOURNAL = {Proc. A.},
FJOURNAL = {Proceedings A},
VOLUME = {473},
YEAR = {2017},
NUMBER = {2207},
PAGES = {20170104, 16},
ISSN = {1364-5021},
MRCLASS = {60H10 (34F05)},
MRNUMBER = {3741822},
MRREVIEWER = {Daniel C. Biles},
DOI = {10.1098/rspa.2017.0104},
URL = {https://doi.org/10.1098/rspa.2017.0104},
}
@article {QiWang2017,
AUTHOR = {Qi, Ruisheng and Wang, Xiaojie},
TITLE = {An accelerated exponential time integrator for semi-linear
stochastic strongly damped wave equation with additive noise},
JOURNAL = {J. Math. Anal. Appl.},
FJOURNAL = {Journal of Mathematical Analysis and Applications},
VOLUME = {447},
YEAR = {2017},
NUMBER = {2},
PAGES = {988--1008},
ISSN = {0022-247X},
MRCLASS = {35R60 (35A35 35L35 35L76)},
MRNUMBER = {3573128},
DOI = {10.1016/j.jmaa.2016.09.052},
URL = {https://doi.org/10.1016/j.jmaa.2016.09.052},
}
@article {MullerGronbachYaroslavtseva2018,
AUTHOR = {M\"{u}ller-Gronbach, Thomas and Yaroslavtseva, Larisa},
TITLE = {A note on strong approximation of {SDE}s with smooth
coefficients that have at most linearly growing derivatives},
JOURNAL = {J. Math. Anal. Appl.},
FJOURNAL = {Journal of Mathematical Analysis and Applications},
VOLUME = {467},
YEAR = {2018},
NUMBER = {2},
PAGES = {1013--1031},
ISSN = {0022-247X},
MRCLASS = {60H10 (35R60 60H35)},
MRNUMBER = {3842417},
DOI = {10.1016/j.jmaa.2018.07.041},
URL = {https://doi.org/10.1016/j.jmaa.2018.07.041},
}
@Article{DaPratoJentzenRoeckner_Mild,
Title = {A mild {I}t\^{o} formula for {SPDE}s},
Author = {Giuseppe {Da Prato} and Arnulf Jentzen and Michael R{\"o}ckner},
Journal = {arXiv:1009.3526},
Year = {2010},
Pages = {39 pages},
Note = {To appear in \textit{Trans.~Amer.~Math.~Soc.}}
}
@book{Hytonen2016analysis,
AUTHOR={Hyt{\"o}nen, Tuomas and van Neerven, Jan and Veraar, Mark and Weis, Lutz},
TITLE={Analysis in Banach Spaces: Volume I: Martingales and Littlewood-Paley Theory},
PUBLISHER={Springer},
YEAR={2016}
}
@Article{GerencserJentzenSalimova2017,
Title = {On stochastic differential equations with arbitrarily slow convergence rates for strong approximation in two space dimensions},
Author = {{Gerencs\'er}, M. and {Jentzen}, A. and {Salimova}, D. },
Journal = {arXiv:1702.03229},
Year = {2017},
Pages = {25 pages},
}
@Article{MullerGronbachYaroslavtseva2017,
author = {Thomas M{\"u}ller-Gronbach and Larisa Yaroslavtseva},
title = {On sub-polynomial lower error bounds for quadrature of {SDE}s with bounded smooth coefficients},
journal = {Stochastic Analysis and Applications},
volume = {0},
number = {0},
pages = {1-29},
year = {2017},
doi = {10.1080/07362994.2016.1263157}
}
@article{HutzenthalerJentzenKruseNguyen2020,
title={{Multilevel Picard approximations for high-dimensional semilinear second-order PDEs with Lipschitz nonlinearities}},
author={Martin Hutzenthaler and Arnulf Jentzen and Thomas Kruse and Tuan Anh Nguyen},
journal={arXiv:2009.02484},
year={2020},
pages={37 pages},
}
@article{GrohsHerrmann2020,
title={Deep neural network approximation for high-dimensional elliptic {PDE}s with boundary conditions},
author={Philipp Grohs and Lukas Herrmann},
year={2020},
journal={arXiv:2007.05384},
archivePrefix={arXiv},
primaryClass={math.NA},
pages = {22 pages}
}
@article{HornungJentzenSalimova2020,
title={Space-time deep neural network approximations for high-dimensional partial differential equations},
author={Fabian Hornung and Arnulf Jentzen and Diyora Salimova},
year={2020},
journal={arXiv:2006.02199},
archivePrefix={arXiv},
pages={52 pages},
primaryClass={math.PR}
}
@Article{Yaroslavtseva2016,
Title = {On non-polynomial lower error bounds for adaptive strong approximation of {SDE}s},
Author = {{Yaroslavtseva}, L.},
Journal = {arXiv:1609.08073},
Year = {2016},
Pages = {19 pages},
}
@Article{MullerGronbachYaroslavtseva2016,
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Note = {To appear}
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Volume = {106},
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PAGES = {565--617},
ISSN = {2194-0401},
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year={2015},
publisher={Pergamon}
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Year = {2013},
Pages = {84 pages},
Adsnote = {Provided by the SAO/NASA Astrophysics Data System},
Archiveprefix = {arXiv},
Eprint = {1309.5595},
Keywords = {Mathematics - Probability, 60H10, 60H15, 34F05},
Primaryclass = {math.PR}
}
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@book {Klenke2014,
AUTHOR = {Klenke, Achim},
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EDITION = {German},
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note = {An introduction with applications},
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groups = {sdiyora:6},
mrclass = {60H10 (60G44 60J60)},
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edition = {Second},
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Title = {On numerical approximation of stochastic {B}urgers' equation},
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Booktitle = {From stochastic calculus to mathematical finance},
Publisher = {Springer},
Year = {2006},
Address = {Berlin},
Pages = {1--15},
Mrclass = {60H35 (60H15 65M06)},
Mrnumber = {MR2233532 (2007c:60064)},
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year = {1991},
pages = {347--386},
volume = {89},
fjournal = {Probability Theory and Related Fields},
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mrclass = {46B20 (46-01)},
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mrclass = {60J60 (65C99 91B28)},
mrnumber = {2186814 (2006h:60130)},
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Title = {Finite element and difference approximation of some linear
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Year = {1998},
Number = {1-2},
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Volume = {64},
Coden = {STOCBS},
Fjournal = {Stochastics and Stochastics Reports},
ISSN = {1045-1129},
Mrclass = {60H15 (65U05)},
Mrnumber = {MR1637047 (99d:60067)}
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groups = {sdiyora:6},
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mrnumber = {1353558 (96h:60052)},
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owner = {scox},
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Title = {Theoretical and numerical aspects of stochastic nonlinear
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Author = {de Bouard, Anne and Debussche, Arnaud and Di Menza, Laurent},
Booktitle = {Journ\'ees ``\'{E}quations aux {D}\'eriv\'ees {P}artielles''
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Publisher = {Univ. Nantes},
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Pages = {Exp. No. III, 13},
Mrclass = {60H15 (35B40 35R60 65C30)},
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Mrreviewer = {Hongjun Gao}
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timestamp = {2016.02.02},
}
@Article{Chow2009,
author = {P.-L Chow and A. Lang and J. Potthoff},
journal = {Preprint, urn:nbn:de:bsz:180-madoc-23291},
title = {Almost sure convergence of a semidiscrete {M}ilstein scheme for {SPDE}'s of {Z}akai type},
year = {2009},
fjournal = {Preprint},
groups = {sdiyora:6},
owner = {sdiyora},
timestamp = {2016.02.02},
}
@InCollection{Clark1980,
author = {Clark, J. M. C. and Cameron, R. J.},
booktitle = {Stochastic differential systems ({P}roc. {IFIP}-{WG} 7/1 {W}orking {C}onf., {V}ilnius, 1978)},
publisher = {Springer},
title = {The maximum rate of convergence of discrete approximations for stochastic differential equations},
year = {1980},
address = {Berlin},
pages = {162--171},
series = {Lecture Notes in Control and Information Sci.},
volume = {25},
groups = {sdiyora:6},
mrclass = {60H10},
mrnumber = {609181 (82f:60133)},
mrreviewer = {G. N. Mil{\cprime}shte{\u\i}n},
owner = {sdiyora},
timestamp = {2016.02.02},
}
@Article{Cliffe2000,
author = {Cliffe, K. A. and Graham, I. G. and Scheichl, R. and Stals, L.},
journal = {J. Comput. Phys.},
title = {Parallel computation of flow in heterogeneous media modelled by mixed finite elements},
year = {2000},
issn = {0021-9991},
number = {2},
pages = {258--282},
volume = {164},
coden = {JCTPAH},
doi = {10.1006/jcph.2000.6593},
fjournal = {Journal of Computational Physics},
groups = {sdiyora:6},
mrclass = {76M10},
mrnumber = {1792513 (2001f:76059)},
owner = {sdiyora},
timestamp = {2016.02.02},
url = {http://dx.doi.org/10.1006/jcph.2000.6593},
}
@Article{Cohen2011,
author = {David Cohen and Magdalena Sigg},
journal = {Numer. Math.},
title = {Convergence analysis of trigonometric methods for stiff second-order stochastic differential equations},
year = {2011},
doi = {10.1007/s00211-011-0426-8},
fjournal = {Numerische Mathematik},
groups = {sdiyora:6},
owner = {sdiyora},
timestamp = {2016.02.02},
url = {http://www.springerlink.com/content/ukt7211324175207/},
}
@Article{Cohen2007,
author = {Cohen, Serge and Rosi{\'n}ski, Jan},
journal = {Bernoulli},
title = {Gaussian approximation of multivariate {L}\'evy processes with applications to simulation of tempered stable processes},
year = {2007},
issn = {1350-7265},
number = {1},
pages = {195--210},
volume = {13},
doi = {10.3150/07-BEJ6011},
fjournal = {Bernoulli. Official Journal of the Bernoulli Society for Mathematical Statistics and Probability},
groups = {sdiyora:6},
mrclass = {60G51 (60G52)},
mrnumber = {2307403 (2008a:60120)},
mrreviewer = {Sonia Fourati},
owner = {sdiyora},
timestamp = {2016.02.02},
url = {http://dx.doi.org/10.3150/07-BEJ6011},
}
@Book{c93,
Title = {Measure theory},
Author = {Cohn, Donald L.},
Publisher = {Birkh\"auser Boston Inc.},
Year = {1993},
Address = {Boston, MA},
Note = {Reprint of the 1980 original},
ISBN = {0-8176-3003-1},
Mrclass = {28-01},
Mrnumber = {MR1454121 (98b:28001)},
Pages = {x+373}
}
@Article{Cont2005,
author = {Cont, Rama},
journal = {International Journal of Theoretical and Applied Finance (IJTAF)},
title = {Modeling Term Structure Dynamics: An Infinite Dimensional Approach},
year = {2005},
number = {03},
pages = {357--380},
volume = {08},
abstract = {Motivated by stylized statistical properties of interest rates, we propose a modeling approach in which the forward rate curve is described as a stochastic process in a space of curves. After decomposing the movements of the term structure into the variations of the short rate, the long rate and the deformation of the curve around its average shape, this deformation is described as the solution of a stochastic evolution equation in an infinite dimensional space of curves. In the case where deformations are local in maturity, this equation reduces to a stochastic PDE, of which we give the simplest example. We discuss the properties of the solutions and show that they capture in a parsimonious manner the essential features of yield curve dynamics: imperfect correlation between maturities, mean reversion of interest rates, the structure of principal components of forward rates and their variances. In particular we show that a flat, constant volatility structures already captures
many of the observed properties. Finally, we discuss parameter estimation issues and show that the model parameters have a natural interpretation in terms of empirically observed quantities.},
groups = {sdiyora:6},
keywords = {Term structure of interest rates; forward rates; multifactor models; Hilbert space; stochastic PDE; random field},
owner = {sdiyora},
timestamp = {2016.02.02},
url = {http://EconPapers.repec.org/RePEc:wsi:ijtafx:v:08:y:2005:i:03:p:357-380},
}
@Book{Cont2004,
author = {Cont, Rama and Tankov, Peter},
publisher = {Chapman \& Hall/CRC, Boca Raton, FL},
title = {Financial modelling with jump processes},
year = {2004},
isbn = {1-5848-8413-4},
note = {535 pages},
series = {Chapman \& Hall/CRC Financial Mathematics Series},
groups = {sdiyora:6},
mrclass = {91-02 (60J75 91B28 91B84)},
mrnumber = {2042661 (2004m:91004)},
owner = {sdiyora},
pages = {xvi+535},
timestamp = {2016.02.02},
}
@Book{Conus2008a,
author = {Daniel Conus},
publisher = {{\'E}cole {P}olytechnique {F}{\'e}d{\'e}derale {D}e {L}ausanne},
title = {The Non-linear {S}tochastic {W}ave {E}quation in {H}igh {D}imensions: {E}xistence, {H}{\"o}lder-continuity and It{\^o}-Taylor {E}xpansion},
year = {2008},
note = {Dissertation},
groups = {sdiyora:6},
owner = {sdiyora},
timestamp = {2016.02.02},
}
@Article{Conus2008,
author = {Daniel Conus and Robert C.~Dalang},
journal = {Electronic Journal of Probability},
title = {The non-linear stochastic wave equation in high dimensions. Electronic Journal of Probability},
year = {2008},
pages = {629-670},
volume = {13},
fjournal = {Electronic Journal of Probability},
groups = {sdiyora:6},
owner = {sdiyora},
timestamp = {2016.02.02},
}
@Article{Conus2014,
author = {Daniel Conus and Arnulf Jentzen and Ryan Kurniawan},
journal = {working paper},
title = {{W}eak convergence rates of spectral {G}alerkin approximations for {SPDE}s with nonlinear diffusion coefficients},
year = {2014},
pages = {26 pages},
groups = {sdiyora:6},
owner = {scox},
timestamp = {2014.07.25},
}
@Article{Conus2014a,
author = {Daniel Conus and Arnulf Jentzen and Ryan Kurniawan},
journal = {working paper},
title = {Weak convergence rates of spectral {G}alerkin approximations for stochastic evolution equations with nonlinear diffusion coefficients},
year = {2014},
pages = {XX pages},
groups = {sdiyora:6},
owner = {sdiyora},
timestamp = {2016.02.02},
}
@Article{Conus2014b,
author = {Daniel Conus and Arnulf Jentzen and Ryan Kurniawan},
journal = {arXiv:1408.1108},
title = {Weak convergence rates of spectral {G}alerkin approximations for {SPDE}s with nonlinear diffusion coefficients},
year = {2014},
pages = {1--29},
groups = {sdiyora:6},
owner = {sdiyora},
timestamp = {2016.02.02},
}
@Article{Cook1970,
author = {H. Cook},
journal = {Acta Metallurgica},
title = {Brownian motion in spinodal decomposition},
year = {1970},
number = {3},
pages = {297--306},
volume = {18},
fjournal = {Acta Metallurgica},
groups = {sdiyora:6},
owner = {sdiyora},
timestamp = {2016.02.02},
}
@Article{Cox1985,
author = {John C. Cox and Jr. Jonathan E. Ingersoll and Stephen A. Ross},
journal = {Econometrica},
title = {A theory of the term structure of interest rates},
year = {1985},
issn = {0012-9682},
number = {2},
pages = {385--407},
volume = {53},
coden = {ECMTA7},
doi = {10.2307/1911242},
fjournal = {Econometrica. Journal of the Econometric Society},
groups = {sdiyora:6},
mrclass = {90A14},
mrnumber = {785475},
owner = {sdiyora},
timestamp = {2016.02.02},
url = {http://dx.doi.org/10.2307/1911242},
}
@Unpublished{Cox2014,
author = {Sonja Cox and Martin Hutzenthaler and Arnulf Jentzen and Jan van Neerven},
note = {Unpublished manuscript},
title = {Convergence in the H\"{o}lder norm with applications to Monte Carlo methods in infinite dimensions},
month = March,
year = {2014},
groups = {sdiyora:6},
owner = {sdiyora},
timestamp = {2016.02.02},
}
@Article{cn13,
Title = {Pathwise {H}{\"o}lder convergence of the implicit-linear {E}uler scheme for semi-linear {SPDE}s with multiplicative noise},
Author = {Cox, Sonja and Neerven, Jan},
Journal = {Numerische Mathematik},
Year = {2013},
Pages = {1-87},
Doi = {10.1007/s00211-013-0538-4},
ISSN = {0029-599X},
Keywords = {Primary 65C30; Secondary 60H15; 60H35; 65J08},
Language = {English},
Publisher = {Springer-Verlag}
}
@Article{Cox2013,
author = {Cox, Sonja and van Neerven, Jan},
journal = {Numer. Math.},
title = {Pathwise {H}\"older convergence of the implicit-linear {E}uler scheme for semi-linear {SPDE}s with multiplicative noise},
year = {2013},
issn = {0029-599X},
number = {2},
pages = {259--345},
volume = {125},
coden = {NUMMA7},
doi = {10.1007/s00211-013-0538-4},
fjournal = {Numerische Mathematik},
groups = {sdiyora:6},
mrclass = {Preliminary Data},
mrnumber = {3101829},
owner = {sdiyora},
timestamp = {2016.02.02},
url = {http://dx.doi.org/10.1007/s00211-013-0538-4},
}
@Article{cvn13,
Title = {Pathwise {H}\"older convergence of the implicit-linear {E}uler
scheme for semi-linear {SPDE}s with multiplicative noise},
Author = {Cox, Sonja and van Neerven, Jan},
Journal = {Numer. Math.},
Year = {2013},
Number = {2},
Pages = {259--345},
Volume = {125},
Doi = {10.1007/s00211-013-0538-4},
Fjournal = {Numerische Mathematik},
ISSN = {0029-599X},
Mrclass = {65C30 (60H15 60H35 65J08)},
Mrnumber = {3101829},
Mrreviewer = {Edward J. Allen},
Url = {http://dx.doi.org/10.1007/s00211-013-0538-4}
}
@Article{cn10,
Title = {Convergence rates of the splitting scheme for parabolic linear
stochastic {C}auchy problems},
Author = {Cox, Sonja and van Neerven, Jan},
Journal = {SIAM J. Numer. Anal.},
Year = {2010},
Number = {2},
Pages = {428--451},
Volume = {48},
Doi = {10.1137/090761835},
Fjournal = {SIAM Journal on Numerical Analysis},
ISSN = {0036-1429},
Mrclass = {65C30 (35R20 35R60 47D06 60J35)},
Mrnumber = {2646103 (2011e:65012)},
Mrreviewer = {Madalina Deaconu}
}
@Article{Cox2010,
author = {Cox, Sonja and van Neerven, Jan},
journal = {SIAM J. Numer. Anal.},
title = {Convergence rates of the splitting scheme for parabolic linear stochastic {C}auchy problems},
year = {2010},
issn = {0036-1429},
number = {2},
pages = {428--451},
volume = {48},
doi = {10.1137/090761835},
fjournal = {SIAM Journal on Numerical Analysis},
groups = {sdiyora:6},
mrclass = {65C30 (35R20 35R60 47D06 60J35)},
mrnumber = {2646103 (2011e:65012)},
mrreviewer = {Madalina Deaconu},
owner = {sdiyora},
timestamp = {2016.02.02},
url = {http://dx.doi.org/10.1137/090761835},
}
@Article{cvn10,
Title = {Convergence rates of the splitting scheme for parabolic linear
stochastic {C}auchy problems},
Author = {Cox, Sonja and van Neerven, Jan},
Journal = {SIAM J. Numer. Anal.},
Year = {2010},
Number = {2},
Pages = {428--451},
Volume = {48},
Doi = {10.1137/090761835},
Fjournal = {SIAM Journal on Numerical Analysis},
ISSN = {0036-1429},
Mrclass = {65C30 (35R20 35R60 47D06 60J35)},
Mrnumber = {2646103 (2011e:65012)},
Mrreviewer = {Madalina Deaconu},
Url = {http://dx.doi.org/10.1137/090761835}
}
@Article{Cox2012,
author = {Sonja G. Cox},
journal = {Ph{D} thesis, TU Delft},
title = {Stochastic {D}ifferential {E}quations in {B}anach {S}paces},
year = {2012},
pages = {239 pages},
groups = {sdiyora:6},
owner = {sdiyora},
timestamp = {2016.02.02},
}
@Article{Cox2012a,
author = {Sonja G. Cox and Erika Hausenblas},
journal = {arXiv:1203.1606},
title = {A perturbation result for quasi-linear stochastic differential equations in {UMD} {B}anach spaces},
year = {2012},
pages = {29 pages},
groups = {sdiyora:6},
owner = {sdiyora},
timestamp = {2016.02.02},
}
@Article{Crepey2013,
author = {Cr\'{e}pey, St\'{e}phane and and Gerboud, R\'{e}mi, and Grbac, Zorana and Ngor, Nathalie},
journal = {International Journal of Theoretical and Applied Finance},
title = {Counterparty risk and funding: the four wings of the {TVA}},
year = {2013},
number = {02},
pages = {1350006},
volume = {16},
doi = {10.1142/S0219024913500064},
eprint = {http://www.worldscientific.com/doi/pdf/10.1142/S0219024913500064},
groups = {sdiyora:6},
owner = {sdiyora},
timestamp = {2016.02.02},
url = {http://www.worldscientific.com/doi/abs/10.1142/S0219024913500064},
}
@Book{Cramer1999,
author = {Cram{\'e}r, Harald},
publisher = {Princeton University Press},
title = {Mathematical methods of statistics},
year = {1999},
address = {Princeton, NJ},
isbn = {0-691-00547-8},
note = {Reprint of the 1946 original},
series = {Princeton Landmarks in Mathematics},
groups = {sdiyora:6},
mrclass = {62-02},
mrnumber = {1816288 (2001k:62002)},
owner = {sdiyora},
pages = {xvi+575},
timestamp = {2016.02.02},
}
@Article{Crandall1992,
author = {Crandall, Michael G. and Ishii, Hitoshi and Lions, Pierre-Louis},
journal = {Bull. Amer. Math. Soc. (N.S.)},
title = {User's guide to viscosity solutions of second order partial differential equations},
year = {1992},
issn = {0273-0979},
number = {1},
pages = {1--67},
volume = {27},
coden = {BAMOAD},
doi = {10.1090/S0273-0979-1992-00266-5},
fjournal = {American Mathematical Society. Bulletin. New Series},
groups = {sdiyora:6},
mrclass = {35J60 (35B05 35D05 35G20)},
mrnumber = {1118699 (92j:35050)},
mrreviewer = {P. Szeptycki},
owner = {sdiyora},
timestamp = {2016.02.02},
url = {http://dx.doi.org/10.1090/S0273-0979-1992-00266-5},
}
@Article{Crandall1983,
author = {Crandall, Michael G. and Lions, Pierre-Louis},
journal = {Trans. Amer. Math. Soc.},
title = {Viscosity solutions of {H}amilton-{J}acobi equations},
year = {1983},
issn = {0002-9947},
number = {1},
pages = {1--42},
volume = {277},
coden = {TAMTAM},
doi = {10.2307/1999343},
fjournal = {Transactions of the American Mathematical Society},
groups = {sdiyora:6},
mrclass = {35F20},
mrnumber = {690039 (85g:35029)},
mrreviewer = {Moshe Marcus},
owner = {sdiyora},
timestamp = {2016.02.02},
url = {http://dx.doi.org/10.2307/1999343},
}
@Article{Crandall1981,
author = {Crandall, Michael G. and Lions, Pierre-Louis},
journal = {C. R. Acad. Sci. Paris S\'er. I Math.},
title = {Condition d'unicit\'e pour les solutions g\'en\'eralis\'ees des \'equations de {H}amilton-{J}acobi du premier ordre},
year = {1981},
issn = {0151-0509},
number = {3},
pages = {183--186},
volume = {292},
coden = {CRSMDY},
fjournal = {Comptes Rendus des S\'eances de l'Acad\'emie des Sciences. S\'erie I. Math\'ematique},
groups = {sdiyora:6},
mrclass = {49C10},
mrnumber = {610314 (82c:49020)},
owner = {sdiyora},
timestamp = {2016.02.02},
}
@Article{Creutzig2009,
author = {Jakob Creutzig and Steffen Dereich and Thomas M{\"u}ller-Gronbach and Klaus Ritter},
journal = {Found. Comput. Math.},
title = {Infinite-dimensional quadrature and approximation of distributions},
year = {2009},
issn = {1615-3375},
number = {4},
pages = {391--429},
volume = {9},
doi = {10.1007/s10208-008-9029-x},
fjournal = {Foundations of Computational Mathematics. The Journal of the Society for the Foundations of Computational Mathematics},
groups = {sdiyora:6},
mrclass = {65D30 (60J60 62E17 65C05)},
mrnumber = {2519865 (2010h:65027)},
mrreviewer = {Michael J. Evans},
owner = {sdiyora},
timestamp = {2016.02.02},
url = {http://dx.doi.org/10.1007/s10208-008-9029-x},
}
@Article{c06,
Title = {Particle approximations for a class of stochastic partial
differential equations},
Author = {Crisan, D.},
Journal = {Appl. Math. Optim.},
Year = {2006},
Number = {3},
Pages = {293--314},
Volume = {54},
Coden = {AMOMBN},
Fjournal = {Applied Mathematics and Optimization. An International Journal
with Applications to Stochastics},
ISSN = {0095-4616},
Mrclass = {60H15 (60G35 62M20 93E11)},
Mrnumber = {MR2268660 (2007m:60174)},
Mrreviewer = {A. F. Gualtierotti}
}
@Article{Cris2006,
author = {D. Crisan},
journal = {Appl. Math. Optim.},
title = {Particle approximations for a class of stochastic partial differential equations},
year = {2006},
issn = {0095-4616},
number = {3},
pages = {293--314},
volume = {54},
coden = {AMOMBN},
fjournal = {Applied Mathematics and Optimization. An International Journal with Applications to Stochastics},
groups = {sdiyora:6},
mrclass = {60H15 (60G35 62M20 93E11)},
mrnumber = {MR2268660 (2007m:60174)},
mrreviewer = {A. F. Gualtierotti},
owner = {sdiyora},
timestamp = {2016.02.02},
}
@Article{c03,
Title = {Exact rates of convergence for a branching particle
approximation to the solution of the {Z}akai equation},
Author = {Crisan, Dan},
Journal = {Ann. Probab.},
Year = {2003},
Number = {2},
Pages = {693--718},
Volume = {31},
Coden = {APBYAE},
Fjournal = {The Annals of Probability},
ISSN = {0091-1798},
Mrclass = {60H35 (60G07 60G35 60H15 60H30 93E11)},
Mrnumber = {MR1964946 (2004i:60099)},
Mrreviewer = {A. F. Gualtierotti}
}
@Article{Cris2003,
author = {Dan Crisan},
journal = {Ann. Probab.},
title = {Exact rates of convergence for a branching particle approximation to the solution of the {Z}akai equation},
year = {2003},
issn = {0091-1798},
number = {2},
pages = {693--718},
volume = {31},
coden = {APBYAE},
fjournal = {The Annals of Probability},
groups = {sdiyora:6},
mrclass = {60H35 (60G07 60G35 60H15 60H30 93E11)},
mrnumber = {MR1964946 (2004i:60099)},
mrreviewer = {A. F. Gualtierotti},
owner = {sdiyora},
timestamp = {2016.02.02},
}
@Article{cgl98,
Title = {Convergence of a branching particle method to the solution of
the {Z}akai equation},
Author = {Crisan, Dan and Gaines, Jessica and Lyons, Terry},
Journal = {SIAM J. Appl. Math.},
Year = {1998},
Number = {5},
Pages = {1568--1590 (electronic)},
Volume = {58},
Fjournal = {SIAM Journal on Applied Mathematics},
ISSN = {0036-1399},
Mrclass = {60G57 (65U05 93E11)},
Mrnumber = {MR1637870 (99e:60117)},
Mrreviewer = {A. Ya. Olenko}
}
@Article{cl99,
Title = {A particle approximation of the solution of the
{K}ushner-{S}tratonovitch equation},
Author = {Crisan, D. and Lyons, T.},
Journal = {Probab. Theory Related Fields},
Year = {1999},
Number = {4},
Pages = {549--578},
Volume = {115},
Coden = {PTRFEU},
Fjournal = {Probability Theory and Related Fields},
ISSN = {0178-8051},
Mrclass = {93E11 (60G46 65C30 65C60)},
Mrnumber = {MR1728921 (2000m:93102)},
Mrreviewer = {P. Neumann}
}
@Article{Crisan1999,
author = {D. Crisan and T. Lyons},
journal = {Probab. Theory Related Fields},
title = {A particle approximation of the solution of the {K}ushner-{S}tratonovitch equation},
year = {1999},
issn = {0178-8051},
number = {4},
pages = {549--578},
volume = {115},
coden = {PTRFEU},
fjournal = {Probability Theory and Related Fields},
groups = {sdiyora:6},
mrclass = {93E11 (60G46 65C30 65C60)},
mrnumber = {MR1728921 (2000m:93102)},
mrreviewer = {P. Neumann},
owner = {sdiyora},
timestamp = {2016.02.02},
}
@Article{Crisan2012,
author = {Crisan, D and Manolarakis, K},
journal = {SIAM Journal on Financial Mathematics},
title = {Solving backward stochastic differential equations using the cubature method: Application to nonlinear pricing},
year = {2012},
number = {1},
pages = {534--571},
volume = {3},
groups = {sdiyora:6},
owner = {sdiyora},
publisher = {SIAM},
timestamp = {2016.02.02},
}
@Article{Crisan2010,
author = {Crisan, Dan and Manolarakis, Konstantinos},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
title = {Probabilistic methods for semilinear partial differential equations. {A}pplications to finance},
year = {2010},
number = {05},
pages = {1107--1133},
volume = {44},
groups = {sdiyora:6},
owner = {sdiyora},
publisher = {Cambridge Univ Press},
timestamp = {2016.02.02},
}
@Article{Crisan2007a,
author = {Crisan, DO and Manolarakis, Konstantinos},
journal = {preprint},
title = {Numerical solution for a BSDE using the Cubature method},
year = {2007},
groups = {sdiyora:6},
owner = {martin},
timestamp = {2014.07.25},
}
@Article{Crisan2010b,
author = {Crisan, Dan and Manolarakis, Konstantinos and Touzi, Nizar},
journal = {Stochastic Processes and their Applications},
title = {On the {M}onte {C}arlo simulation of {BSDEs}: An improvement on the {M}alliavin weights},
year = {2010},
number = {7},
pages = {1133--1158},
volume = {120},
groups = {sdiyora:6},
owner = {sdiyora},
publisher = {Elsevier},
timestamp = {2016.02.02},
}
@InCollection{cx07,
Title = {Numerical solutions for a class of {SPDE}s over bounded
domains},
Author = {Crisan, Dan and Xiong, Jie},
Booktitle = {Conference {O}xford sur les m\'ethodes de {M}onte {C}arlo
s\'equentielles},
Publisher = {EDP Sci., Les Ulis},
Year = {2007},
Pages = {121--125},
Series = {ESAIM Proc.},
Volume = {19},
Mrclass = {60H35 (60H15 65C30)},
Mrnumber = {MR2405657 (2009g:60093)}
}
@Article{Curry2014,
author = {Curry, Charles and Ebrahimi-Fard, Kurusch and Malham, Simon J. A. and Wiese, Anke},
journal = {Stochastics},
title = {L\'evy processes and quasi-shuffle algebras},
year = {2014},
issn = {1744-2508},
number = {4},
pages = {632--642},
volume = {86},
doi = {10.1080/17442508.2013.865131},
fjournal = {Stochastics. An International Journal of Probability and Stochastic Processes},
groups = {sdiyora:6},
mrclass = {60H30 (16T30 60G44 60G51)},
mrnumber = {3230071},
owner = {sdiyora},
timestamp = {2016.02.02},
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mrreviewer = {Ya. {\=I}. B{\={\i}}lopol{\cprime}s{\cprime}ka},
owner = {sdiyora},
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timestamp = {2016.02.02},
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}
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mrnumber = {2276524 (2008a:60153)},
mrreviewer = {Wilhelm Stannat},
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timestamp = {2016.02.02},
url = {http://dx.doi.org/10.1007/s11118-006-9021-5},
}
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mrclass = {60H15 (35Q30 35R60 76D05 76M35)},
mrnumber = {2005200 (2004m:60133)},
mrreviewer = {Marek Capi{\'n}ski},
owner = {sdiyora},
timestamp = {2016.02.02},
url = {http://dx.doi.org/10.1016/S0021-7824(03)00025-4},
}
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urldate = {2013-01-21},
}
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AUTHOR = {Gy\"ongy, Istv\'an},
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mrnumber = {MR2465711},
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Title = {Rate of convergence of implicit approximations for stochastic
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}
@InCollection{Gyoengy2007,
author = {Istv{\'a}n Gy{\"o}ngy and Annie Millet},
booktitle = {Stochastic differential equations: theory and applications},
publisher = {World Sci. Publ., Hackensack, NJ},
title = {Rate of convergence of implicit approximations for stochastic evolution equations},
year = {2007},
pages = {281--310},
series = {Interdiscip. Math. Sci.},
volume = {2},
groups = {sdiyora:6},
mrclass = {60H15 (60H35)},
mrnumber = {MR2393581 (2009d:60207)},
mrreviewer = {Carlos M. Mora Gonz{\'a}lez},
owner = {sdiyora},
timestamp = {2016.02.02},
}
@Article{gm05,
Title = {On discretization schemes for stochastic evolution equations},
Author = {Gy{\"o}ngy, Istv{\'a}n and Millet, Annie},
Journal = {Potential Anal.},
Year = {2005},
Number = {2},
Pages = {99--134},
Volume = {23},
Coden = {POANE2},
Fjournal = {Potential Analysis. An International Journal Devoted to the
Interactions between Potential Theory, Probability Theory,
Geometry and Functional Analysis},
ISSN = {0926-2601},
Mrclass = {60H15 (65C30)},
Mrnumber = {MR2139212 (2006a:60115)},
Mrreviewer = {Carlos M. Mora Gonz{\'a}lez}
}
@Article{gn97,
Title = {Implicit scheme for stochastic parabolic partial differential
equations driven by space-time white noise},
Author = {Gy{\"o}ngy, Istv{\'a}n and Nualart, David},
Journal = {Potential Anal.},
Year = {1997},
Number = {4},
Pages = {725--757},
Volume = {7},
Coden = {POANE2},
Fjournal = {Potential Analysis. An International Journal Devoted to the
Interactions between Potential Theory, Probability Theory,
Geometry and Functional Analysis},
ISSN = {0926-2601},
Mrclass = {60H15 (35K60 35R60 65M06)},
Mrnumber = {MR1480861 (98m:60097)},
Mrreviewer = {Ralf Manthey}
}
@Article{Gyoengy1997,
author = {Istv{\'a}n Gy{\"o}ngy and David Nualart},
journal = {Potential Anal.},
title = {Implicit scheme for stochastic parabolic partial differential equations driven by space-time white noise},
year = {1997},
issn = {0926-2601},
number = {4},
pages = {725--757},
volume = {7},
coden = {POANE2},
fjournal = {Potential Analysis. An International Journal Devoted to the Interactions between Potential Theory, Probability Theory, Geometry and Functional Analysis},
groups = {sdiyora:6},
mrclass = {60H15 (35K60 35R60 65M06)},
mrnumber = {MR1480861 (98m:60097)},
mrreviewer = {Ralf Manthey},
owner = {sdiyora},
timestamp = {2016.02.02},
}
@Article{gn95,
Title = {Implicit scheme for quasi-linear parabolic partial
differential equations perturbed by space-time white noise},
Author = {Gy{\"o}ngy, Istv{\'a}n and Nualart, David},
Journal = {Stochastic Process. Appl.},
Year = {1995},
Number = {1},
Pages = {57--72},
Volume = {58},
Coden = {STOPB7},
Fjournal = {Stochastic Processes and their Applications},
ISSN = {0304-4149},
Mrclass = {65M12 (35R60 60H15 65U05)},
Mrnumber = {MR1341554 (96b:65091)}
}
@Article{Gyoengy1995,
author = {Istv{\'a}n Gy{\"o}ngy and David Nualart},
journal = {Stochastic Process. Appl.},
title = {Implicit scheme for quasi-linear parabolic partial differential equations perturbed by space-time white noise},
year = {1995},
issn = {0304-4149},
number = {1},
pages = {57--72},
volume = {58},
coden = {STOPB7},
fjournal = {Stochastic Processes and their Applications},
groups = {sdiyora:6},
mrclass = {65M12 (35R60 60H15 65U05)},
mrnumber = {MR1341554 (96b:65091)},
owner = {sdiyora},
timestamp = {2016.02.02},
}
@Article{Gyoengy2011,
author = {Gy{\"o}ngy, Istv{\'a}n and R{\'a}sonyi, Mikl{\'o}s},
journal = {Stochastic Process. Appl.},
title = {A note on {E}uler approximations for {SDE}s with {H}\"older continuous diffusion coefficients},
year = {2011},
issn = {0304-4149},
number = {10},
pages = {2189--2200},
volume = {121},
coden = {STOPB7},
doi = {10.1016/j.spa.2011.06.008},
fjournal = {Stochastic Processes and their Applications},
groups = {sdiyora:6},
mrclass = {60H35 (60F15)},
mrnumber = {2822773},
owner = {sdiyora},
timestamp = {2016.02.02},
url = {http://dx.doi.org/10.1016/j.spa.2011.06.008},
}
@Article{GoengySabanisS2014,
Title = {{Convergence of tamed {E}uler schemes for a class of stochastic evolution equations}},
Author = {{Gy{\"o}ngy}, I. and {Sabanis}, S. and {{\v S}i{\v s}ka}, D. },
Journal = {arXiv:1308.1796},
Year = {2015},
Pages = {17 pages}
}
@Article{GoengySabanisS2015,
author="Gy{\"o}ngy, Istv{\'a}n
and Sabanis, Sotirios
and {\v{S}}i{\v{s}}ka, David",
title="Convergence of tamed {E}uler schemes for a class of stochastic evolution equations",
journal="Stoch. Partial Differ. Equ. Anal. Comput.",
year="2015",
pages="1--21",
abstract="We prove stability and convergence of a full discretization for a class of stochastic evolution equations with super-linearly growing operators appearing in the drift term. This is done by using the recently developed tamed Euler method, which employs a fully explicit time stepping, coupled with a Galerkin scheme for the spatial discretization.",
issn="2194-041X",
doi="10.1007/s40072-015-0057-7",
url="http://dx.doi.org/10.1007/s40072-015-0057-7"
}
@Article{gs06,
Title = {Rate of convergence of {W}ong-{Z}akai approximations for
stochastic partial differential equations},
Author = {Gy{\"o}ngy, Istv{\'a}n and Shmatkov, Anton},
Journal = {Appl. Math. Optim.},
Year = {2006},
Number = {3},
Pages = {315--341},
Volume = {54},
Coden = {AMOMBN},
Fjournal = {Applied Mathematics and Optimization. An International Journal
with Applications to Stochastics},
ISSN = {0095-4616},
Mrclass = {60H35 (35R60 60H15)},
Mrnumber = {MR2268661 (2008e:60212)},
Mrreviewer = {Carlos M. Mora Gonz{\'a}lez}
}
@Article{Gyoengy2006c,
author = {Istv{\'a}n Gy{\"o}ngy and Anton Shmatkov},
journal = {Appl. Math. Optim.},
title = {Rate of convergence of {W}ong-{Z}akai approximations for stochastic partial differential equations},
year = {2006},
issn = {0095-4616},
number = {3},
pages = {315--341},
volume = {54},
coden = {AMOMBN},
fjournal = {Applied Mathematics and Optimization. An International Journal with Applications to Stochastics},
groups = {sdiyora:6},
mrclass = {60H35 (35R60 60H15)},
mrnumber = {MR2268661 (2008e:60212)},
mrreviewer = {Carlos M. Mora Gonz{\'a}lez},
owner = {sdiyora},
timestamp = {2016.02.02},
}
@Book{Hoermander1990,
author = {H{\"o}rmander, Lars},
publisher = {Springer-Verlag},
title = {The analysis of linear partial differential operators. {I}: Distribution theory and {F}ourier analysis},
year = {1990},
address = {Berlin},
edition = {Second},
isbn = {3-540-52343-X},
series = {Springer Study Edition},
groups = {sdiyora:6},
mrclass = {35-02 (42B10 46Fxx)},
mrnumber = {1065136 (91m:35001b)},
owner = {sdiyora},
pages = {xii+440},
timestamp = {2016.02.02},
}
@Article{Hoermander1967,
author = {H{\"o}rmander, Lars},
journal = {Acta Math.},
title = {Hypoelliptic second order differential equations},
year = {1967},
issn = {0001-5962},
pages = {147--171},
volume = {119},
fjournal = {Acta Mathematica},
groups = {sdiyora:6},
mrclass = {35.48 (47.00)},
mrnumber = {0222474 (36 \$5526)},
mrreviewer = {J. Smoller},
owner = {sdiyora},
timestamp = {2016.02.02},
}
@Book{hnw93,
Title = {Solving ordinary differential equations. {I}},
Author = {Hairer, E. and N{\o}rsett, S. P. and Wanner, G.},
Publisher = {Springer-Verlag},
Year = {1993},
Address = {Berlin},
Edition = {Second},
Note = {Nonstiff problems},
Series = {Springer Series in Computational Mathematics},
Volume = {8},
ISBN = {3-540-56670-8},
Mrclass = {65-02 (34A50 65L99)},
Mrnumber = {MR1227985 (94c:65005)},
Pages = {xvi+528}
}
@Article{Hairer,
author = {Martin Hairer},
journal = {To appear in Ann. of Math. (2)},
title = {Solving the {KPZ} equation},
pages = {103 pages},
fjournal = {To appear in Annals of Mathematics. Second Series},
groups = {sdiyora:6},
owner = {sdiyora},
timestamp = {2016.02.02},
}
@Article{Hairer2013,
author = {Hairer, Martin},
journal = {arXiv:1303.5113},
title = {A theory of regularity structures},
year = {2013},
pages = {178 pages},
groups = {sdiyora:6},
owner = {sdiyora},
timestamp = {2016.02.02},
}
@Article{Hairer2011,
author = {Hairer, Martin},
journal = {Bull. Sci. Math.},
title = {On {M}alliavin's proof of {H}\"ormander's theorem},
year = {2011},
issn = {0007-4497},
number = {6-7},
pages = {650--666},
volume = {135},
coden = {BSMQA9},
doi = {10.1016/j.bulsci.2011.07.007},
fjournal = {Bulletin des Sciences Math\'ematiques},
groups = {sdiyora:6},
mrclass = {60H07 (60H30 60J60)},
mrnumber = {2838095},
owner = {sdiyora},
timestamp = {2016.02.02},
url = {http://dx.doi.org/10.1016/j.bulsci.2011.07.007},
}
@Article{Hairer2011a,
author = {Martin Hairer},
journal = {arXiv:1109.6811},
title = {Solving the {KPZ} equation},
year = {2011},
pages = {102 pages},
groups = {sdiyora:6},
owner = {sdiyora},
timestamp = {2016.02.02},
}
@Article{Hairer2009,
author = {Martin Hairer},
journal = {arXiv:0907.4178v1},
title = {{A}n {I}ntroduction to {S}tochastic {PDE}s},
year = {2009},
pages = {78 pages},
groups = {sdiyora:6},
owner = {sdiyora},
timestamp = {2016.02.02},
}
@Article{Hairer2008,
author = {Hairer, M.},
journal = {Unpublished lecture notes},
title = {Ergodic theory for Stochastic PDEs},
year = {2008},
groups = {sdiyora:6},
owner = {sdiyora},
timestamp = {2016.02.02},
}
@article {Hairer2015,
AUTHOR = {Hairer, Martin and Hutzenthaler, Martin and Jentzen, Arnulf},
TITLE = {Loss of regularity for {K}olmogorov equations},
JOURNAL = {Ann. Probab.},
FJOURNAL = {The Annals of Probability},
VOLUME = {43},
YEAR = {2015},
NUMBER = {2},
PAGES = {468--527},
ISSN = {0091-1798},
MRCLASS = {35K15 (35B65)},
MRNUMBER = {3305998},
DOI = {10.1214/13-AOP838},
URL = {http://dx.doi.org/10.1214/13-AOP838},
}
@Article{Hairer2014,
author = {Martin Hairer and Martin Hutzenthaler and Arnulf Jentzen},
journal = {To appear in Ann. Probab.},
title = {Loss of regularity for {K}olmogorov equations},
year = {2014},
pages = {arXiv:1209.6035},
groups = {sdiyora:6},
owner = {sdiyora},
timestamp = {2016.02.02},
}
@Article{Hairer2010,
author = {Martin Hairer and Jan Maas},
journal = {arXiv:1011.0966v1},
title = {A spatial version of the {I}to-{S}tratonovich correction},
year = {2010},
pages = {33 pages},
groups = {sdiyora:6},
owner = {sdiyora},
timestamp = {2016.02.02},
}
@Article{Hairer2011b,
author = {Hairer, Martin and Mattingly, Jonathan C.},
journal = {Electron. J. Probab.},
title = {A theory of hypoellipticity and unique ergodicity for semilinear stochastic {PDE}s},
year = {2011},
issn = {1083-6489},
pages = {no. 23, 658--738},
volume = {16},
doi = {10.1214/EJP.v16-875},
fjournal = {Electronic Journal of Probability},
groups = {sdiyora:6},
mrclass = {60H15 (35H10 35R60 60H07)},
mrnumber = {2786645 (2012e:60170)},
mrreviewer = {Feng-Yu Wang},
owner = {sdiyora},
timestamp = {2016.02.02},
url = {http://dx.doi.org/10.1214/EJP.v16-875},
}
@Article{Hairer2006,
author = {Hairer, Martin and Mattingly, Jonathan C.},
journal = {Ann. of Math. (2)},
title = {Ergodicity of the 2{D} {N}avier-{S}tokes equations with degenerate stochastic forcing},
year = {2006},
issn = {0003-486X},
number = {3},
pages = {993--1032},
volume = {164},
coden = {ANMAAH},
doi = {10.4007/annals.2006.164.993},
fjournal = {Annals of Mathematics. Second Series},
groups = {sdiyora:6},
mrclass = {37L55 (35Q30 35R60 60H15 76D05 76M35)},
mrnumber = {2259251 (2008a:37095)},
mrreviewer = {Hakima Bessaih},
owner = {sdiyora},
timestamp = {2016.02.02},
url = {http://dx.doi.org/10.4007/annals.2006.164.993},
}
@Article{Hairer2012,
author = {Martin Hairer and Marc D. Ryser and Hendrik Weber},
journal = {Electron. J. Probab.},
title = {Triviality of the 2{D} stochastic {A}llen-{C}ahn equation},
year = {2012},
number = {39},
pages = {1--14},
volume = {17},
fjournal = {Electronic Journal of Probability},
groups = {sdiyora:6},
owner = {sdiyora},
timestamp = {2016.02.02},
}
@InCollection{Hairer2009a,
author = {Martin Hairer and Andrew Stuart and Jochen Vo{\ss}},
booktitle = {Trends in stochastic analysis},
publisher = {Cambridge Univ. Press},
title = {Sampling conditioned diffusions},
year = {2009},
address = {Cambridge},
pages = {159--185},
series = {London Math. Soc. Lecture Note Ser.},
volume = {353},
groups = {sdiyora:6},
mrclass = {60-08 (60J22 60J60 62M20 65C30)},
mrnumber = {2562154},
owner = {sdiyora},
timestamp = {2016.02.02},
}
@Article{Hairera,
author = {M. Hairer and A. M. Stuart and J. Voss},
journal = {To appear in Ann. Appl. Probab.},
title = {Sampling Conditioned Hypoelliptic Diffusions},
pages = {22 pages},
fjournal = {To appear in The Annals of Applied Probability},
groups = {sdiyora:6},
owner = {sdiyora},
timestamp = {2016.02.02},
}
@Article{Hairer2007,
author = {M. Hairer and A. M. Stuart and J. Voss},
journal = {Ann. Appl. Probab.},
title = {Analysis of {SPDE}s arising in path sampling. {II}. {T}he nonlinear case},
year = {2007},
issn = {1050-5164},
number = {5-6},
pages = {1657--1706},
volume = {17},
doi = {10.1214/07-AAP441},
fjournal = {The Annals of Applied Probability},
groups = {sdiyora:6},
mrclass = {60H15 (60G17 60G35 65C30)},
mrnumber = {2358638 (2008i:60105)},
mrreviewer = {Jan I. Seidler},
owner = {sdiyora},
timestamp = {2016.02.02},
url = {http://dx.doi.org/10.1214/07-AAP441},
}
@Article{Hairer2005,
author = {M. Hairer and A. M. Stuart and J. Voss and P. Wiberg},
journal = {Commun. Math. Sci.},
title = {Analysis of {SPDE}s arising in path sampling. {I}. {T}he {G}aussian case},
year = {2005},
issn = {1539-6746},
number = {4},
pages = {587--603},
volume = {3},
fjournal = {Communications in Mathematical Sciences},
groups = {sdiyora:6},
mrclass = {60H15 (60G15 60G35 60H10)},
mrnumber = {2188686 (2006i:60086)},
mrreviewer = {Krystyna Twardowska},
owner = {sdiyora},
timestamp = {2016.02.02},
url = {http://projecteuclid.org/getRecord?id=euclid.cms/1144429334},
}
@Article{Hairer2010a,
author = {Martin Hairer and Jochen Voss},
journal = {arXiv:1005.4438v1},
title = {Approximations to the Stochastic {B}urgers Equation},
year = {2010},
pages = {19 pages},
groups = {sdiyora:6},
owner = {sdiyora},
timestamp = {2016.02.02},
}
@Article{Haken1985,
author = {Haken, H. and Kelso, J. A. S. and Bunz, H.},
journal = {Biol. Cybernet.},
title = {A theoretical model of phase transitions in human hand movements},
year = {1985},
issn = {0340-1200},
number = {5},
pages = {347--356},
volume = {51},
coden = {BICYAF},
doi = {10.1007/BF00336922},
fjournal = {Biological Cybernetics},
groups = {sdiyora:6},
mrclass = {92A09 (82A25)},
mrnumber = {778098 (86d:92007)},
owner = {sdiyora},
timestamp = {2016.02.02},
url = {http://dx.doi.org/10.1007/BF00336922},
}
@Article{Halidias2015,
Title = {{Construction of positivity preserving numerical schemes for some multidimensional stochastic differential equations}},
Author = {Nikolaos Halidias},
Journal = {{Discrete Contin. Dyn. Syst. Ser. B}},
Year = {2015},
Number = {1},
Pages = {153-160},
Volume = {20},
Doi = {10.3934/dcdsb.2015.20.153},
ISSN = {1531-3492},
Url = {http://aimsciences.org/journals/displayArticlesnew.jsp?paperID=10508}
}
@Article{Halidias2013,
Title = {A novel approach to construct numerical methods for stochastic differential equations},
Author = {Halidias, Nikolaos},
Journal = {Numer. Algorithms},
Year = {2013},
Number = {1},
Pages = {79--87},
Volume = {66},
Abstract = {In this paper we propose a new numerical method for solving stochastic differential equations (SDEs). As an application of this method we propose an explicit numerical scheme for a super linear SDE for which the usual Euler scheme diverges.},
Doi = {10.1007/s11075-013-9724-9},
ISSN = {1572-9265},
Url = {http://dx.doi.org/10.1007/s11075-013-9724-9}
}
@Article{Halidias2012,
author = {Halidias, Nikolaos},
journal = {Int. J. Comput. Math.},
title = {Semi-discrete approximations for stochastic differential equations and applications},
year = {2012},
issn = {0020-7160},
number = {6},
pages = {780--794},
volume = {89},
doi = {10.1080/00207160.2012.658380},
fjournal = {International Journal of Computer Mathematics},
groups = {sdiyora:6},
mrclass = {65C30 (60H35)},
mrnumber = {2903590},
owner = {sdiyora},
timestamp = {2016.02.02},
url = {http://dx.doi.org/10.1080/00207160.2012.658380},
}
@Article{Halidias2008,
author = {Halidias, Nikolaos and Kloeden, Peter E.},
journal = {BIT},
title = {A note on the {E}uler-{M}aruyama scheme for stochastic differential equations with a discontinuous monotone drift coefficient},
year = {2008},
issn = {0006-3835},
number = {1},
pages = {51--59},
volume = {48},
doi = {10.1007/s10543-008-0164-1},
fjournal = {BIT. Numerical Mathematics},
groups = {sdiyora:6},
mrclass = {60H10 (60H20)},
mrnumber = {2386114 (2009c:60150)},
mrreviewer = {Renato G. C. Spigler},
owner = {sdiyora},
timestamp = {2016.02.02},
url = {http://dx.doi.org/10.1007/s10543-008-0164-1},
}
@Article{HalidiasStamatiou2013,
Title = {{On the numerical solution of some nonlinear stochastic differential equations using the semi-discrete method}},
Author = {{Halidias}, N. and {Stamatiou}, I.~S.},
Journal = {arXiv:1309.3189},
Year = {2014},
Pages = {36 pages}
}
@Book{h76,
Title = {Naive {M}engenlehre},
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Note = {Vierte Auflage,
Aus dem Englischen {\"u}bersetzt von Manfred Armbrust und
Fritz Ostermann,
Moderne Mathematik in elementarer Darstellung, No. 6},
Mrclass = {04-01},
Mrnumber = {MR0427066 (55 \#102)},
Pages = {132}
}
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Title = {Strong stability and non-smooth data error estimates for
discretizations of linear parabolic problems},
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Year = {2002},
Number = {2},
Pages = {351--379},
Volume = {42},
Doi = {10.1023/A:1021903109720},
Fjournal = {BIT. Numerical Mathematics},
ISSN = {0006-3835},
Mrclass = {65M12 (65M15)},
Mrnumber = {1912592 (2003d:65085)}
}
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author = {Hardy, G. H. and Littlewood, J. E. and P{\'o}lya, G.},
publisher = {Cambridge University Press, Cambridge},
title = {Inequalities},
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isbn = {0-521-35880-9},
note = {Reprint of the 1952 edition},
series = {Cambridge Mathematical Library},
groups = {sdiyora:6},
mrclass = {26Dxx (01A75)},
mrnumber = {944909 (89d:26016)},
owner = {sdiyora},
pages = {xii+324},
timestamp = {2016.02.02},
}
@Article{h07a,
Title = {Finite element approximation of stochastic partial
differential equations driven by {P}oisson random measures of
jump type},
Author = {Hausenblas, Erika},
Journal = {SIAM J. Numer. Anal.},
Year = {2007/08},
Number = {1},
Pages = {437--471},
Volume = {46},
Fjournal = {SIAM Journal on Numerical Analysis},
ISSN = {0036-1429},
Mrclass = {60H35 (60H15 65C30)},
Mrnumber = {MR2377271 (2009g:60094)},
Mrreviewer = {Fran{\c{c}}ois Delarue}
}
@Article{Hausenblas2007/08a,
author = {Erika Hausenblas},
journal = {SIAM J. Numer. Anal.},
title = {Finite element approximation of stochastic partial differential equations driven by {P}oisson random measures of jump type},
year = {2007/08},
issn = {0036-1429},
number = {1},
pages = {437--471},
volume = {46},
fjournal = {SIAM Journal on Numerical Analysis},
groups = {sdiyora:6},
mrclass = {60H35 (60H15 65C30)},
mrnumber = {MR2377271 (2009g:60094)},
mrreviewer = {Fran\c{c}ois Delarue},
owner = {sdiyora},
timestamp = {2016.02.02},
}
@Article{Hausenblas2010,
author = {Erika Hausenblas},
journal = {J. Comput. Appl. Math.},
title = {Weak approximation of the stochastic wave equation},
year = {2010},
number = {2},
pages = {33--58},
volume = {235},
fjournal = {Journal of Computational and Applied Mathematics},
groups = {sdiyora:6},
owner = {sdiyora},
timestamp = {2016.02.02},
}
@Article{h07b,
Title = {Wong-{Z}akai type approximation of {SPDE}s of {L}\'evy noise},
Author = {Hausenblas, Erika},
Journal = {Acta Appl. Math.},
Year = {2007},
Number = {2},
Pages = {99--134},
Volume = {98},
Coden = {AAMADV},
Fjournal = {Acta Applicandae Mathematicae. An International Survey Journal
on Applying Mathematics and Mathematical Applications},
ISSN = {0167-8019},
Mrclass = {60H15 (60B12 60G55)},
Mrnumber = {MR2336465 (2008g:60192)},
Mrreviewer = {Nicholas Frangos}
}
@Article{h03a,
Title = {Approximation for semilinear stochastic evolution equations},
Author = {Hausenblas, Erika},
Journal = {Potential Anal.},
Year = {2003},
Number = {2},
Pages = {141--186},
Volume = {18},
Coden = {POANE2},
Fjournal = {Potential Analysis. An International Journal Devoted to the
Interactions between Potential Theory, Probability Theory,
Geometry and Functional Analysis},
ISSN = {0926-2601},
Mrclass = {60H15 (65C30)},
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}
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owner = {sdiyora},
timestamp = {2016.02.02},
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Title = {Stochastic {P}artial {D}ifferential {E}quations:
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Author = {Arnulf Jentzen},
Note = {Lecture notes, ETH Zurich, summer semester 2015, available online at \url{https://people.math.ethz.ch/~grsam/NASPDE_MATH_FS15/script/NASPDE_130.pdf}},
Month = {September},
Year = {2015},
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Timestamp = {2015.09.14},
Url = {http://www.math.ethz.ch/education/bachelor/lectures/fs2015/math/numsol}
}
@Unpublished{Jentzen2018DNNlecturenotes,
Title = {{A Mathematical Introduction to Machine
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Author = {Arnulf Jentzen},
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}
@Unpublished{Jentzen2014SPDElecturenotes,
Title = {Numerical Analysis of Stochastic Partial Differential Equations},
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Note = {Lecture notes, ETH Zurich, summer semester 2014, available online at http://www.math.ethz.ch/education/bachelor/lectures/fs2014/math/numsol},
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Timestamp = {2014.07.25},
Url = {http://www.math.ethz.ch/education/bachelor/lectures/fs2014/math/numsol}
}
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@Book{j09b,
Title = {Taylor Expansions
for Stochastic Partial
Differential Equations},
Author = {Jentzen, Arnulf},
Publisher = {Johann Wolfgang Goethe University},
Year = {2009},
Address = {Frankfurt am Main, Germany},
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@Book{Jentzen2009a,
author = {Arnulf Jentzen},
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title = {Taylor Expansions for Stochastic Partial Differential Equations},
year = {2009},
address = {Frankfurt am Main, Germany},
note = {Dissertation},
groups = {sdiyora:6},
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timestamp = {2016.02.02},
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Title = {Numerische Verfahren
hoher Ordnung f{\"u}r
zuf{\"a}llige Differentialgleichungen},
Author = {Jentzen, Arnulf},
Publisher = {Johann Wolfgang Goethe University},
Year = {2007},
Address = {Frankfurt am Main, Germany},
Note = {Diplomarbeit}
}
@Book{Jentzen2007,
author = {Arnulf Jentzen},
publisher = {Johann Wolfgang Goethe University},
title = {Numerische Verfahren hoher Ordnung f{\"u}r zuf{\"a}llige Differentialgleichungen},
year = {2007},
address = {Frankfurt am Main, Germany},
note = {Diplomarbeit},
groups = {sdiyora:6},
owner = {sdiyora},
timestamp = {2016.02.02},
}
@Book{Jentzen2012,
author = {Jentzen, Arnulf and Kloeden, Peter},
publisher = {Society for Industrial and Applied Mathematics (SIAM)},
title = {{T}aylor {A}pproximations for {S}tochastic {P}artial {D}ifferential {E}quations},
year = {2012},
address = {Philadelphia, PA},
isbn = {978-1-611972-00-9},
series = {CBMS-NSF Regional Conference Series in Applied Mathematics},
volume = {83},
groups = {sdiyora:6},
owner = {sdiyora},
pages = {xvi+220},
timestamp = {2016.02.02},
}
@Article{jkw09,
Title = {{Efficient simulation of nonlinear parabolic SPDEs with additive
noise.}},
Author = {Jentzen, Arnulf and Kloeden, Peter and Winkel, Georg},
Journal = {Ann. Appl. Probab. },
Year = {2011},
Number = {3},
Pages = {908-950},
Volume = {21},
Language = {English}
}
@Book{JentzenKloeden2011,
Title = {Taylor approximations for stochastic partial differential equations},
Author = {Jentzen, Arnulf and Kloeden, Peter E.},
Publisher = {Society for Industrial and Applied Mathematics (SIAM)},
Year = {2011},
Address = {Philadelphia, PA},
Series = {CBMS-NSF Regional Conference Series in Applied Mathematics},
Volume = {83},
Doi = {10.1137/1.9781611972016},
ISBN = {978-1-611972-00-9},
Mrclass = {60H15 (60H35 65C30)},
Mrnumber = {2856611},
Mrreviewer = {Sergey V. Lototsky},
Pages = {xiv+220},
Url = {http://dx.doi.org/10.1137/1.9781611972016}
}
@Article{jk09c,
Title = {Taylor expansions of solutions of stochastic partial differential equations with additive noise},
Author = {Jentzen, Arnulf and Kloeden, Peter E.},
Journal = {Ann. Probab.},
Year = {2010},
Number = {2},
Pages = {532--569},
Volume = {38},
Fjournal = {The Annals of Probability}
}
@Article{Jentzen2009c,
author = {Arnulf Jentzen and Peter E. Kloeden},
journal = {Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci.},
title = {Overcoming the order barrier in the numerical approximation of stochastic partial differential equations with additive space-time noise},
year = {2009},
issn = {1364-5021},
number = {2102},
pages = {649--667},
volume = {465},
fjournal = {Proceedings of The Royal Society of London. Series A. Mathematical, Physical and Engineering Sciences},
groups = {sdiyora:6},
mrclass = {65C30 (60H15 60H35)},
mrnumber = {MR2471778},
owner = {sdiyora},
timestamp = {2016.02.02},
}
@Article{jk09a,
Title = {Pathwise {T}aylor schemes for random ordinary differential
equations},
Author = {Jentzen, Arnulf and Kloeden, Peter E.},
Journal = {BIT},
Year = {2009},
Number = {1},
Pages = {113--140},
Volume = {49},
Fjournal = {BIT. Numerical Mathematics},
ISSN = {0006-3835},
Mrclass = {65C30},
Mrnumber = {MR2486131}
}
@Article{jk09d,
Title = {The numerical approximation of stochastic partial differential equations},
Author = {Jentzen, Arnulf and Kloeden, Peter E.},
Journal = {Milan J. Math.},
Year = {2009},
Number = {1},
Pages = {205--244},
Volume = {77},
Fjournal = {Milan Journal of Mathematics},
ISSN = {1424-9286}
}
@Article{jk09e,
Title = {A unified existence and uniqueness theorem for stochastic evolution equations},
Author = {Jentzen, Arnulf and Kloeden, Peter E.},
Journal = {Bull. Austral. Math. Soc.},
Year = {2009},
Note = {To appear},
Pages = {17 pages},
Fjournal = {Bulletin of the Australian Mathematical Society}
}
@Article{Jentzen2007a,
author = {Arnulf Jentzen and Peter E. Kloeden},
journal = {Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci.},
title = {Pathwise convergent higher order numerical schemes for random ordinary differential equations},
year = {2007},
issn = {1364-5021},
number = {2087},
pages = {2929--2944},
volume = {463},
fjournal = {Proceedings of The Royal Society of London. Series A. Mathematical, Physical and Engineering Sciences},
groups = {sdiyora:6},
mrclass = {65C30 (65L07)},
mrnumber = {MR2360184 (2008m:65017)},
mrreviewer = {Domingo Hern{\'a}ndez-Abreu},
owner = {sdiyora},
timestamp = {2016.02.02},
}
@Article{Jentzen2009e,
author = {A. Jentzen and P. E. Kloeden and A. Neuenkirch},
journal = {Numer. Math.},
title = {Pathwise approximation of stochastic differential equations on domains: higher order convergence rates without global {L}ipschitz coefficients},
year = {2009},
issn = {0029-599X},
number = {1},
pages = {41--64},
volume = {112},
coden = {NUMMA7},
fjournal = {Numerische Mathematik},
groups = {sdiyora:6},
mrclass = {60H35},
mrnumber = {MR2481529},
owner = {sdiyora},
timestamp = {2016.02.02},
}
@InCollection{Jentzen2009f,
author = {A. Jentzen and P. E. Kloeden and A. Neuenkirch},
booktitle = {Foundations of computational mathematics, {H}ong {K}ong 2008},
publisher = {Cambridge Univ. Press},
title = {Pathwise convergence of numerical schemes for random and stochastic differential equations},
year = {2009},
address = {Cambridge},
pages = {140--161},
series = {London Math. Soc. Lecture Note Ser.},
volume = {363},
groups = {sdiyora:6},
mrclass = {65C30 (60H10 60H35)},
mrnumber = {2562499},
owner = {sdiyora},
timestamp = {2016.02.02},
}
@Article{jkn09a,
Title = {Pathwise approximation of stochastic differential equations on
domains: higher order convergence rates without global
{L}ipschitz coefficients},
Author = {Jentzen, A. and Kloeden, P. E. and Neuenkirch, A.},
Journal = {Numer. Math.},
Year = {2009},
Number = {1},
Pages = {41--64},
Volume = {112},
Coden = {NUMMA7},
Fjournal = {Numerische Mathematik},
ISSN = {0029-599X},
Mrclass = {60H35},
Mrnumber = {MR2481529}
}
@InCollection{jkn09b,
Title = {Pathwise convergence of numerical schemes for random and stochastic differential equations},
Author = {Jentzen, A. and Kloeden, P. E. and Neuenkirch, A.},
Publisher = {Cambridge University Press},
Year = {2009},
Pages = {140--161},
Editors = {Cucker, F. and Pinkus, A. and Todd, M.},
Fjournal = {Foundations of Computational Mathematics: Hong Kong 2008},
Journal = {Found. Comput. Math.}
}
@Article{Jentzen2011b,
author = {Arnulf Jentzen and Peter E. Kloeden and Georg Winkel},
journal = {Ann. Appl. Probab.},
title = {Efficient simulation of nonlinear parabolic {SPDE}s with additive noise},
year = {2011},
number = {3},
pages = {908--950},
volume = {21},
fjournal = {The Annals of Applied Probability},
groups = {sdiyora:6},
owner = {sdiyora},
timestamp = {2016.02.02},
}
@Article{JentzenKurniawan2015,
Title = {{Weak convergence rates for Euler-type approximations of semilinear stochastic evolution equations with nonlinear diffusion coefficients}},
Author = {{Jentzen}, A. and {Kurniawan}, R.},
Journal = {arXiv:1501.03539},
Year = {2015},
Pages = {51 pages}
}
@Article{jlsbs09,
Title = {An improved Maximum Allowable Transfer Interval for Networked Control Systems},
Author = {Jentzen, A. and Leber, F. and Schneisgen, D. and Berger, A. and Siegmund, S},
Journal = {IEEE Trans. Automat. Control},
Year = {2009},
Note = {in revision},
Pages = {10},
Fjournal = {Institute of Electrical and Electronics Engineers.
Transactions on Automatic Control}
}
@Article{Jentzen2015b,
author = {Arnulf Jentzen and Thomas M{\"u}ller-Gronbach and Larisa Yaroslavtseva},
journal = {arXiv:1506.02828},
title = {On stochastic differential equations with arbitrary slow convergence rates for strong approximation},
year = {2015},
pages = {1--26},
groups = {sdiyora:6},
owner = {sdiyora},
timestamp = {2016.02.02},
}
@Article{jn09,
Title = {A random {E}uler scheme for {C}arath\'eodory differential
equations},
Author = {Jentzen, A. and Neuenkirch, A.},
Journal = {J. Comput. Appl. Math.},
Year = {2009},
Number = {1},
Pages = {346--359},
Volume = {224},
Coden = {JCAMDI},
Fjournal = {Journal of Computational and Applied Mathematics},
ISSN = {0377-0427},
Mrclass = {65L05},
Mrnumber = {MR2474237}
}
@Article{jp2015,
Title = {{Strong convergence rates for an explicit numerical approximation method for stochastic evolution equations with non-globally Lipschitz continuous nonlinearities}},
Author = {{Jentzen}, A. and {Pu{\v s}nik}, P.},
Journal = {arXiv:1504.03523},
Year = {2015},
Pages = {38 pages},
Note = {To appear in \emph{IMA J.\ Num.\ Anal.}}
}
@Article{Jentzen2015c,
author = {Jentzen, Arnulf and R{\"o}ckner, Michael},
journal = {Found. Comput. Math.},
title = {A {M}ilstein scheme for {SPDE}s},
year = {2015},
issn = {1615-3375},
number = {2},
pages = {313--362},
volume = {15},
doi = {10.1007/s10208-015-9247-y},
fjournal = {Foundations of Computational Mathematics. The Journal of the Society for the Foundations of Computational Mathematics},
groups = {sdiyora:6},
mrclass = {65C30 (60H35)},
mrnumber = {3320928},
owner = {sdiyora},
timestamp = {2016.02.02},
url = {http://dx.doi.org/10.1007/s10208-015-9247-y},
}
@Article{jr10,
Title = {Regularity analysis of stochastic partial differential equations with nonlinear multiplicative trace class noise},
Author = {Jentzen, Arnulf and R{\"o}ckner, Michael},
Journal = {Journal of Differential Equations},
Year = {2012},
Number = {1},
Pages = {114--136},
Volume = {252},
Fjournal = {Journal of Differential Equations}
}
@Article{jr10b,
Title = {A {M}ilstein scheme for {SPDEs}},
Author = {Jentzen, Arnulf and R{\"o}ckner, Michael},
Journal = {arXiv:1001.2751v4},
Year = {2012},
Pages = {37 pages}
}
@InCollection{j91,
Title = {Lattice approximation of a nonlinear stochastic partial
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Author = {Jetschke, Gottfried},
Booktitle = {Random partial differential equations ({O}berwolfach, 1989)},
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Pages = {107--126},
Series = {Internat. Ser. Numer. Math.},
Volume = {102},
Mrclass = {60H15 (35R60)},
Mrnumber = {MR1185742 (93h:60091)},
Mrreviewer = {Wilfried Grecksch}
}
@InCollection{Jetschke1991,
author = {Gottfried Jetschke},
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publisher = {Birkh{\"a}user},
title = {Lattice approximation of a nonlinear stochastic partial differential equation with white noise},
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address = {Basel},
pages = {107--126},
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volume = {102},
groups = {sdiyora:6},
mrclass = {60H15 (35R60)},
mrnumber = {MR1185742 (93h:60091)},
mrreviewer = {Wilfried Grecksch},
owner = {sdiyora},
timestamp = {2016.02.02},
}
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owner = {sdiyora},
timestamp = {2016.02.02},
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}
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timestamp = {2016.02.02},
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timestamp = {2016.02.02},
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timestamp = {2016.02.02},
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year = {2013},
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groups = {sdiyora:6},
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timestamp = {2016.02.02},
}
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mrclass = {60J65 (35K99 35R60 60G44 60H10 60J60)},
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timestamp = {2016.02.02},
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Title = {Noise regularization and computations for the 1-dimensional
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Mrclass = {35R60 (35B25 35K55 65C30)},
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mrreviewer = {Pedro Mar{\'i}n Rubio},
owner = {sdiyora},
timestamp = {2016.02.02},
url = {http://dx.doi.org/10.1214/105051605000000511},
}
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@Book{k06,
Title = {Wahrscheinlichkeitstheorie},
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}
@Article{kj07,
Title = {Pathwise convergent higher order numerical schemes for random
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Mrclass = {65C30 (65L07)},
Mrnumber = {MR2360184 (2008m:65017)},
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}
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timestamp = {2016.02.02},
url = {http://dx.doi.org/10.1016/j.cam.2010.08.011},
}
@Article{klns09,
Title = {The exponential integrator scheme for stochastic partial differential equations: pathwise error bounds},
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Year = {2009},
Note = {Submitted (2009)}
}
@Article{Kloeden2013,
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year = {2013},
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@article {KovacsLarssonLindgren2018,
AUTHOR = {Kov\'{a}cs, Mih\'{a}ly and Larsson, Stig and Lindgren, Fredrik},
TITLE = {On the discretisation in time of the stochastic {A}llen-{C}ahn
equation},
JOURNAL = {Math. Nachr.},
FJOURNAL = {Mathematische Nachrichten},
VOLUME = {291},
YEAR = {2018},
NUMBER = {5-6},
PAGES = {966--995},
ISSN = {0025-584X},
MRCLASS = {60H15 (60H35 65M75)},
MRNUMBER = {3795566},
DOI = {10.1002/mana.201600283},
URL = {https://doi.org/10.1002/mana.201600283},
}
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Title = {{On the discretization in time of the stochastic Allen-Cahn equation}},
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Year = {2015},
Pages = {34 pages}
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url = {http://dx.doi.org/10.1239/jap/1437658601},
}
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year = {2013},
pages = {16 pages},
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timestamp = {2016.02.02},
}
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title = {Weak convergence of finite element approximations of linear stochastic evolution equations with additive noise},
year = {2012},
issn = {0006-3835},
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mrreviewer = {B{\"u}lent Karas{\"o}zen},
owner = {sdiyora},
timestamp = {2016.02.02},
url = {http://dx.doi.org/10.1007/s10543-011-0344-2},
}
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Title = {Strong convergence of the finite element method with truncated
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Year = {2010},
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Doi = {10.1007/s11075-009-9281-4},
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author = {Kov{\'a}cs, Mih{\'a}ly and Larsson, Stig and Lindgren, Fredrik},
journal = {Numer. Algorithms},
title = {Strong convergence of the finite element method with truncated noise for semilinear parabolic stochastic equations with additive noise},
year = {2010},
issn = {1017-1398},
number = {2-3},
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volume = {53},
doi = {10.1007/s11075-009-9281-4},
fjournal = {Numerical Algorithms},
groups = {sdiyora:6},
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mrnumber = {2600932},
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owner = {sdiyora},
timestamp = {2016.02.02},
url = {http://dx.doi.org/10.1007/s11075-009-9281-4},
}
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author = {Kov{\'a}cs, Mih{\'a}ly and Larsson, Stig and Mesforush, Ali},
journal = {SIAM J. Numer. Anal.},
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year = {2011},
issn = {0036-1429},
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url = {http://dx.doi.org/10.1137/110828150},
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author = {Mih{\'a}ly Kov{\'a}cs and Stig Larsson and Fardin Saedpanah},
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year = {2010},
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Year = {2010},
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Volume = {48},
Doi = {10.1137/090772241},
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ISSN = {0036-1429},
Mrclass = {65M60 (60H35 65C30)},
Mrnumber = {2646102 (2011c:65215)},
Mrreviewer = {Gon{\c{c}}alo Dos Reis},
Url = {http://dx.doi.org/10.1137/090772241}
}
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Title = {{Consistency and Stability of a Milstein-Galerkin Finite Element Scheme for Semilinear SPDE}},
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Pages = {38 pages},
Keywords = {Mathematics - Numerical Analysis, Mathematics - Probability, 60H15, 65C30, 65M60, 65M70}
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Title = {{Optimal Error Estimates of Galerkin Finite Element Methods for Stochastic Partial Differential Equations with Multiplicative Noise}},
Author = {{Kruse}, R.},
Journal = {arXiv:1103.4504v1},
Year = {2011},
Archiveprefix = {arXiv},
Eprint = {1103.4504},
Keywords = {Mathematics - Numerical Analysis, Mathematics - Analysis of PDEs, Mathematics - Probability, 60H15, 65C30, 65M60, 65M70},
Primaryclass = {math.NA}
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author = {Raphael Kruse},
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@Book{k08,
Title = {Diskrete Approximation
gew{\"o}hnlicher und partieller
stochastischer Differentialgleichungen},
Author = {Kruse, R.},
Publisher = {Univerist{\"a}t Bielefeld},
Year = {2008},
Address = {Bielefeld, Germany},
Note = {Diplomarbeit}
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@Book{Kruse2008,
author = {R. Kruse},
publisher = {Univerist{\"a}t Bielefeld},
title = {Diskrete Approximation gew{\"o}hnlicher und partieller stochastischer Differentialgleichungen},
year = {2008},
address = {Bielefeld, Germany},
note = {Diplomarbeit},
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owner = {sdiyora},
timestamp = {2016.02.02},
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Title = {Optimal regularity for semilinear stochastic partial differential equations with multiplicative noise},
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Volume = {17},
Fjournal = {Electronic Journal of Probability}
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author = {Raphael Kruse and Stig Larsson},
journal = {arXiv:1109.6487v1},
title = {Optimal regularity for semilinear stochastic partial differential equations with multiplicative noise},
year = {2011},
pages = {19 pages},
groups = {sdiyora:6},
owner = {sdiyora},
timestamp = {2016.02.02},
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title = {Optimal regularity for semilinear stochastic partial differential equations with multiplicative noise},
year = {2010},
pages = {17 pages},
groups = {sdiyora:6},
owner = {sdiyora},
timestamp = {2016.02.02},
}
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Publisher = {Dekker},
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url = {http://dx.doi.org/10.1007/BFb0092417},
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Mrreviewer = {Krystyna Twardowska}
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year = {1994},
issn = {0178-8051},
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Volume = {51},
Doi = {10.1137/S0040585X97982487},
Fjournal = {Rossi\u\i skaya Akademiya Nauk. Teoriya Veroyatnoste\u\i\ i ee
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@Book{p83,
Title = {Semigroups of linear operators and applications to partial
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@Book{Pazy1983,
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language = {eng},
owner = {sdiyora},
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timestamp = {2016.02.02},
url = {http://eudml.org/doc/194210},
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author = {Petersdorff, Tobias Von and Schwab, Christoph},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis-Mod{\'e}lisation Math{\'e}matique et Analyse Num{\'e}rique},
title = {Numerical solution of parabolic equations in high dimensions},
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ISSN = {0926-2601},
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Mrclass = {60H35 (60H15)},
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Mrreviewer = {Roger Pettersson}
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year = {2006},
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fjournal = {Potential Analysis. An International Journal Devoted to the Interactions between Potential Theory, Probability Theory, Geometry and Functional Analysis},
groups = {sdiyora:6},
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timestamp = {2016.02.02},
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url = {http://dx.doi.org/10.1137/060673308},
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year = {2006},
issn = {0736-2994},
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volume = {24},
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fjournal = {Stochastic Analysis and Applications},
groups = {sdiyora:6},
mrclass = {65C30 (34F05 37H10 60H10 60H35)},
mrnumber = {2198539 (2006k:65016)},
mrreviewer = {Henri Schurz},
owner = {sdiyora},
timestamp = {2016.02.02},
url = {http://dx.doi.org/10.1080/07362990500397699},
}
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Mrclass = {60H35 (41A58 60H10 60J60 65C30)},
Mrnumber = {MR2095070 (2005m:60151)},
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author = {Andreas R{\"o}{\ss}ler},
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title = {Stochastic {T}aylor expansions for the expectation of functionals of diffusion processes},
year = {2004},
issn = {0736-2994},
number = {6},
pages = {1553--1576},
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coden = {SAAPDA},
doi = {10.1081/SAP-200029495},
fjournal = {Stochastic Analysis and Applications},
groups = {sdiyora:6},
mrclass = {60H35 (41A58 60H10 60J60 65C30)},
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mrreviewer = {G. Leha},
owner = {sdiyora},
timestamp = {2016.02.02},
url = {http://dx.doi.org/10.1081/SAP-200029495},
}
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Title = {Runge-Kutta Methods for
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Publisher = {Shaker Verlag},
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Address = {Aachen},
Note = {Dissertation, Technische
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author = {Andreas R{\"o}{\ss}ler},
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title = {Runge-Kutta Methods for the Numerical Solution of Stochastic Differential Equations},
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address = {Aachen},
note = {Dissertation, Technische Universit{\"a}t Darmstadt, Darmstadt, Germany},
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timestamp = {2016.02.02},
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journal = {Ann. Probab.},
title = {Kolmogorov equations in infinite dimensions: well-posedness and regularity of solutions, with applications to stochastic generalized {B}urgers equations},
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issn = {0091-1798},
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volume = {34},
coden = {APBYAE},
doi = {10.1214/009117905000000666},
fjournal = {The Annals of Probability},
groups = {sdiyora:6},
mrclass = {35R15 (35B30 35Q53 35R60 37L55 60H15 76D06)},
mrnumber = {2223955 (2007b:35323)},
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owner = {sdiyora},
timestamp = {2016.02.02},
url = {http://dx.doi.org/10.1214/009117905000000666},
}
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mrclass = {60H35 (60H10 65C30)},
mrnumber = {2338537 (2009b:60214)},
mrreviewer = {Evelyn Buckwar},
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timestamp = {2016.02.02},
url = {http://dx.doi.org/10.1007/s10543-007-0130-3},
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timestamp = {2016.02.02},
url = {http://dx.doi.org/10.1137/0719041},
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author = {Reed, Michael and Simon, Barry},
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mrclass = {47-02 (81.47)},
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mrnumber = {2334594 (2008e:76170)},
mrreviewer = {Jorge A. Le{\'o}n},
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url = {http://dx.doi.org/10.1016/j.jde.2007.03.027},
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author = {Ren, Jiagang and Zhang, Xicheng},
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mrclass = {60H10 (34A12 34F05)},
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volume = {1},
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mrclass = {65N06 (74N99)},
mrnumber = {1980482 (2004d:65122)},
owner = {sdiyora},
timestamp = {2016.02.02},
url = {http://projecteuclid.org/getRecord?id=euclid.cms/1118152077},
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Title = {An introduction to partial differential equations},
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Edition = {Second},
Series = {Texts in Applied Mathematics},
Volume = {13},
ISBN = {0-387-00444-0},
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Pages = {xiv+434}
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@book {RenardyRogers1993,
AUTHOR = {Renardy, Michael and Rogers, Robert C.},
TITLE = {An introduction to partial differential equations},
SERIES = {Texts in Applied Mathematics},
VOLUME = {13},
PUBLISHER = {Springer-Verlag, New York},
YEAR = {1993},
PAGES = {xiv+428},
ISBN = {0-387-97952-2},
MRCLASS = {35-01 (00A05 46-01 47-01 47F05)},
MRNUMBER = {1211418},
MRREVIEWER = {P. Szeptycki},
}
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year = {1994},
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Title = {Infinite-dimensional dynamical systems},
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title = {Weak approximations of solutions of a first order hyperbolic stochastic partial differential equation},
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volume = {24},
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groups = {sdiyora:6},
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author = {Christian Roth},
publisher = {Martin-Luther-Universit{\"a}t Halle-Wittenberg},
title = {Stochastische partielle Differentialgleichungen 1. Ordnung},
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Pages = {xviii+315}
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note = {Translated from the 1990 Japanese original, Revised edition of the 1999 English translation},
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mrnumber = {3185174},
owner = {sdiyora},
pages = {xiv+521},
timestamp = {2016.02.02},
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title = {Lattice Approximation for Stochastic Reaction Diffusion Equations with One-Sided {L}ipschitz Condition},
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title = {Bounds on sample paths of stochastic nonlinear systems---a {L}yapunov function approach},
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address = {Dordrecht},
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volume = {85},
doi = {10.1007/978-94-010-0886-0_20},
groups = {sdiyora:6},
mrclass = {37H99 (34D45 34F05 60G10)},
mrnumber = {1896845 (2003k:37079)},
mrreviewer = {Henri Schurz},
owner = {sdiyora},
timestamp = {2016.02.02},
url = {http://dx.doi.org/10.1007/978-94-010-0886-0_20},
}
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issn = {0924-090X},
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owner = {sdiyora},
timestamp = {2016.02.02},
url = {http://dx.doi.org/10.1007/BF00120720},
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groups = {sdiyora:6},
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owner = {sdiyora},
timestamp = {2016.02.02},
url = {http://dx.doi.org/10.1007/BF00915273},
}
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mrclass = {34G10 (34D09 34D10 47D06 47N20)},
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mrreviewer = {Ralph Chill},
owner = {scox},
timestamp = {2014.07.25},
}
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author = {Schoutens, Wim},
publisher = {Springer-Verlag, New York},
title = {Stochastic processes and orthogonal polynomials},
year = {2000},
isbn = {0-387-95015-X},
series = {Lecture Notes in Statistics},
volume = {146},
doi = {10.1007/978-1-4612-1170-9},
groups = {sdiyora:6},
mrclass = {60J80 (33C45 42C05 60G51 60J27)},
mrnumber = {1761401 (2001f:60095)},
mrreviewer = {P. R. Parthasarathy},
owner = {sdiyora},
pages = {xiv+163},
timestamp = {2016.02.02},
url = {http://dx.doi.org/10.1007/978-1-4612-1170-9},
}
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Title = {Analysis and discretization of semi-linear stochastic wave
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Journal = {Discrete Contin. Dyn. Syst. Ser. S},
Year = {2008},
Number = {2},
Pages = {353--363},
Volume = {1},
Fjournal = {Discrete and Continuous Dynamical Systems. Series S},
ISSN = {1937-1632},
Mrclass = {60H15 (35L70 35R60 65C30)},
Mrnumber = {MR2379913 (2009b:60201)},
Mrreviewer = {Hisao Watanabe}
}
@Article{Schurz2008,
author = {Henri Schurz},
journal = {Discrete Contin. Dyn. Syst. Ser. S},
title = {Analysis and discretization of semi-linear stochastic wave equations with cubic nonlinearity and additive space-time noise},
year = {2008},
issn = {1937-1632},
number = {2},
pages = {353--363},
volume = {1},
fjournal = {Discrete and Continuous Dynamical Systems. Series S},
groups = {sdiyora:6},
mrclass = {60H15 (35L70 35R60 65C30)},
mrnumber = {MR2379913 (2009b:60201)},
mrreviewer = {Hisao Watanabe},
owner = {sdiyora},
timestamp = {2016.02.02},
}
@Article{s07,
Title = {A numerical method for nonlinear stochastic wave equations in
{$\Bbb R\sp 1$}},
Author = {Schurz, Henri},
Journal = {Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal.},
Year = {2007},
Number = {Advances in Dynamical Systems, suppl. S2},
Pages = {74--78},
Volume = {14},
Coden = {DCDIS},
Fjournal = {Dynamics of Continuous, Discrete \& Impulsive Systems. Series
A. Mathematical Analysis},
ISSN = {1201-3390},
Mrclass = {65C30 (34F05 35R60 60H15 60H35)},
Mrnumber = {MR2384109 (2009d:65014)},
Mrreviewer = {Dror Givon}
}
@Article{Schurz2007,
author = {Henri Schurz},
journal = {Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal.},
title = {A numerical method for nonlinear stochastic wave equations in {$\Bbb R\sp 1$}},
year = {2007},
issn = {1201-3390},
number = {Advances in Dynamical Systems, suppl. S2},
pages = {74--78},
volume = {14},
coden = {DCDIS},
fjournal = {Dynamics of Continuous, Discrete \& Impulsive Systems. Series A. Mathematical Analysis},
groups = {sdiyora:6},
mrclass = {65C30 (34F05 35R60 60H15 60H35)},
mrnumber = {MR2384109 (2009d:65014)},
mrreviewer = {Dror Givon},
owner = {sdiyora},
timestamp = {2016.02.02},
}
@Article{s06,
Title = {Dynamics of linearly partial-implicit midpoint methods for numerical integration of some infinite systems of ODEs with cubic-type nonlinearity and Q-regular additive noise},
Author = {Schurz, Henri},
Journal = {Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal.},
Year = {2006},
Note = {accepted},
Fjournal = {Dynamics of Continuous, Discrete \& Impulsive Systems. Series
A. Mathematical Analysis}
}
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number = {4},
pages = {459--480},
volume = {3},
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groups = {sdiyora:6},
mrclass = {65C30 (60H10)},
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owner = {sdiyora},
timestamp = {2016.02.02},
}
@Article{Schurz2005,
author = {Schurz, Henri},
journal = {Int. J. Numer. Anal. Model.},
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number = {2},
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mrclass = {60H10 (65C30)},
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timestamp = {2016.02.02},
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author = {Schurz, Henri},
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mrclass = {65C30 (60G07 60H10)},
mrnumber = {1988720 (2004e:65014)},
mrreviewer = {Suresh V. Lawande},
owner = {sdiyora},
timestamp = {2016.02.02},
}
@InCollection{Schurz2002,
author = {Schurz, H.},
booktitle = {Handbook of stochastic analysis and applications},
publisher = {Dekker},
title = {Numerical analysis of stochastic differential equations without tears},
year = {2002},
address = {New York},
pages = {237--359},
series = {Statist. Textbooks Monogr.},
volume = {163},
groups = {sdiyora:6},
mrclass = {60H35 (60H10)},
mrnumber = {1882714},
owner = {sdiyora},
timestamp = {2016.02.02},
}
@Article{Schuss1980,
author = {Schuss, Zeev},
journal = {SIAM Review},
title = {Singular Perturbation Methods in Stochastic Differential Equations of Mathematical Physics},
year = {1980},
issn = {00361445},
number = {2},
pages = {119--155},
volume = {22},
abstract = {Stochastic differential equations are used as models for various physical phenomena, such as chemical reactions, atomic migration in crystals, thermal fluctuations in electrical networks, noisy signals in radio transmission, etc. First passage times of solutions of such equations from certain domains and the distribution of the exit points are computed from the solutions of singularly perturbed elliptic boundary value problems. Physical interpretation of these quantities is given. Application in communication theory and in reliability of structures are shown.},
copyright = {Copyright © 1980 Society for Industrial and Applied Mathematics},
groups = {sdiyora:6},
jstor_articletype = {primary_article},
jstor_formatteddate = {Apr., 1980},
owner = {sdiyora},
publisher = {Society for Industrial and Applied Mathematics},
timestamp = {2016.02.02},
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author = {Lorenzo Zambotti},
booktitle = {Stochastic partial differential equations and applications---{VII}},
publisher = {Chapman \& Hall/CRC, Boca Raton, FL},
title = {It{\^o}-{T}anaka's formula for stochastic partial differential equations driven by additive space-time white noise},
year = {2006},
pages = {337--347},
series = {Lect. Notes Pure Appl. Math.},
volume = {245},
groups = {sdiyora:6},
mrclass = {60H15 (35K55 35R60 60H07)},
mrnumber = {MR2227240 (2007g:60072)},
mrreviewer = {Jan I. Seidler},
owner = {sdiyora},
timestamp = {2016.02.02},
}
@Article{Zhang2013,
author = {Zhang, Guannan and Gunzburger, Max and Zhao, Weidong},
journal = {Journal of Computational Mathematics},
title = {A sparse-grid method for multi-dimensional backward stochastic differential equations},
year = {2013},
number = {3},
pages = {221--248},
volume = {31},
groups = {sdiyora:6},
owner = {sdiyora},
publisher = {Oak Ridge National Laboratory (ORNL)},
timestamp = {2016.02.02},
}
@Article{Zhang2004,
author = {Zhang, Jianfeng},
journal = {The Annals of Applied Probability},
title = {A numerical scheme for {BSDEs}},
year = {2004},
number = {1},
pages = {459--488},
volume = {14},
groups = {sdiyora:6},
owner = {sdiyora},
publisher = {Institute of Mathematical Statistics},
timestamp = {2016.02.02},
}
@Article{Zhang2004a,
author = {Qi-Min Zhang and Wei-Guo Zhang and Zan-Kan Nie},
journal = {Appl. Math. Comput.},
title = {Convergence of the {E}uler scheme for stochastic functional partial differential equations},
year = {2004},
issn = {0096-3003},
number = {2},
pages = {479--492},
volume = {155},
coden = {AMHCBQ},
fjournal = {Applied Mathematics and Computation},
groups = {sdiyora:6},
mrclass = {60H15 (65C30)},
mrnumber = {MR2077063 (2005h:60190)},
mrreviewer = {Carlos M. Mora Gonz{\'a}lez},
owner = {sdiyora},
timestamp = {2016.02.02},
}
@Article{zzn04,
Title = {Convergence of the {E}uler scheme for stochastic functional
partial differential equations},
Author = {Zhang, Qi-Min and Zhang, Wei-Guo and Nie, Zan-Kan},
Journal = {Appl. Math. Comput.},
Year = {2004},
Number = {2},
Pages = {479--492},
Volume = {155},
Coden = {AMHCBQ},
Fjournal = {Applied Mathematics and Computation},
ISSN = {0096-3003},
Mrclass = {60H15 (65C30)},
Mrnumber = {MR2077063 (2005h:60190)},
Mrreviewer = {Carlos M. Mora Gonz{\'a}lez}
}
@Article{Zhang2010,
author = {Zhang, Xicheng},
journal = {Stochastic Process. Appl.},
title = {Stochastic flows and {B}ismut formulas for stochastic {H}amiltonian systems},
year = {2010},
issn = {0304-4149},
number = {10},
pages = {1929--1949},
volume = {120},
coden = {STOPB7},
doi = {10.1016/j.spa.2010.05.015},
fjournal = {Stochastic Processes and their Applications},
groups = {sdiyora:6},
mrclass = {60H10 (37H99 60H07 60H20)},
mrnumber = {2673982 (2012c:60160)},
mrreviewer = {Shi Zan Fang},
owner = {sdiyora},
timestamp = {2016.02.02},
url = {http://dx.doi.org/10.1016/j.spa.2010.05.015},
}
@Article{z07,
Title = {Regularities for semilinear stochastic partial differential
equations},
Author = {Zhang, Xicheng},
Journal = {J. Funct. Anal.},
Year = {2007},
Number = {2},
Pages = {454--476},
Volume = {249},
Coden = {JFUAAW},
Doi = {10.1016/j.jfa.2007.03.018},
Fjournal = {Journal of Functional Analysis},
ISSN = {0022-1236},
Mrclass = {60H15 (35R60)},
Mrnumber = {MR2345340 (2008j:60154)},
Mrreviewer = {Alexander M. Kolodi{\u\i}}
}
@Article{Zhang2005,
author = {Zhang, Xicheng},
journal = {Stochastic Process. Appl.},
title = {Homeomorphic flows for multi-dimensional {SDE}s with non-{L}ipschitz coefficients},
year = {2005},
issn = {0304-4149},
number = {3},
pages = {435--448},
volume = {115},
coden = {STOPB7},
doi = {10.1016/j.spa.2004.10.003},
fjournal = {Stochastic Processes and their Applications},
groups = {sdiyora:6},
mrclass = {60H10 (34A12)},
mrnumber = {2118287 (2006b:60132)},
mrreviewer = {Anatoliy A. Levakov},
owner = {sdiyora},
timestamp = {2016.02.02},
url = {http://dx.doi.org/10.1016/j.spa.2004.10.003},
}
@Article{Zhang2014,
Title = {{New explicit balanced schemes for SDEs with locally Lipschitz coefficients}},
Author = {{Zhang}, Z.},
Journal = {arXiv:1402.3708},
Year = {2014},
Pages = {15 pages}
}
@Article{Zhang2014a,
author = {Zhongqiang Zhang},
journal = {arXiv:1402.3708},
title = {New explicit balanced schemes for {SDEs} with locally {L}ipschitz coefficients},
year = {2014},
pages = {1--15},
groups = {sdiyora:6},
owner = {sdiyora},
timestamp = {2016.02.02},
}
@Article{Zhou2010,
author = {Zhou, Xiang and E, Weinan},
journal = {Commun. Math. Sci.},
title = {Study of noise-induced transitions in the {L}orenz system using the minimum action method},
year = {2010},
issn = {1539-6746},
number = {2},
pages = {341--355},
volume = {8},
fjournal = {Communications in Mathematical Sciences},
groups = {sdiyora:6},
mrclass = {37D45 (34C23 34C37 37N20)},
mrnumber = {2664454 (2011j:37062)},
mrreviewer = {Bernd Krauskopf},
owner = {sdiyora},
timestamp = {2016.02.02},
url = {http://projecteuclid.org/getRecord?id=euclid.cms/1274816885},
}
@Article{ZongWuHuang2014,
Title = {{Convergence and stability of the semi-tamed Euler scheme for stochastic differential equations with non-Lipschitz continuous coefficients}},
Author = {Xiaofeng Zong and Fuke Wu and Chengming Huang},
Journal = {Appl. Math. Comput.},
Year = {2014},
Pages = {240 - 250},
Volume = {228},
Doi = {http://dx.doi.org/10.1016/j.amc.2013.11.100},
ISSN = {0096-3003},
Url = {http://www.sciencedirect.com/science/article/pii/S0096300313012691}
}
@book{liu2015stochastic,
title={Stochastic partial differential equations: an introduction},
author={Liu, Wei and R{\"o}ckner, Michael},
year={2015},
publisher={Springer}
}
@Book{Carmona1999,
editor = {Rene A. Carmona and Boris Rozovskii},
publisher = {American Mathematical Society},
title = {Stochastic partial differential equations: six perspectives},
year = {1999},
address = {Providence, RI},
isbn = {0-8218-0806-0},
series = {Mathematical Surveys and Monographs},
volume = {64},
groups = {sdiyora:6},
mrclass = {60-06 (35-06 35R60 60H15)},
mrnumber = {1661761 (99g:60005)},
owner = {sdiyora},
pages = {xii+334},
timestamp = {2016.02.02},
}
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