A Proof that Artificial Neural Networks Overcome the Curse of Dimensionality in the Numerical Approximation of Black–Scholes Partial Differential Equations
Artificial neural networks (ANNs) have very successfully been used in numerical simulations for a series of computational problems ranging from image classification/image recognition, speech recognition, time series analysis, game intelligence, and computational advertising to numerical approximatio...
Uloženo v:
| Hlavní autoři: | , , , |
|---|---|
| Médium: | E-kniha Kniha |
| Jazyk: | angličtina |
| Vydáno: |
Providence, Rhode Island
American Mathematical Society
2023
|
| Vydání: | 1 |
| Edice: | Memoirs of the American Mathematical Society |
| Témata: | |
| ISBN: | 147045632X, 9781470456320 |
| ISSN: | 0065-9266, 1947-6221 |
| On-line přístup: | Získat plný text |
| Tagy: |
Přidat tag
Žádné tagy, Buďte první, kdo vytvoří štítek k tomuto záznamu!
|
| Abstract | Artificial neural networks (ANNs) have very successfully been used in numerical simulations for a series of computational problems
ranging from image classification/image recognition, speech recognition, time series analysis, game intelligence, and computational
advertising to numerical approximations of partial differential equations (PDEs). Such numerical simulations suggest that ANNs have the
capacity to very efficiently approximate high-dimensional functions and, especially, indicate that ANNs seem to admit the fundamental
power to overcome the curse of dimensionality when approximating the high-dimensional functions appearing in the above named
computational problems. There are a series of rigorous mathematical approximation results for ANNs in the scientific literature. Some of
them prove convergence without convergence rates and some of these mathematical results even rigorously establish convergence rates but
there are only a few special cases where mathematical results can rigorously explain the empirical success of ANNs when approximating
high-dimensional functions. The key contribution of this article is to disclose that ANNs can efficiently approximate high-dimensional
functions in the case of numerical approximations of Black-Scholes PDEs. More precisely, this work reveals that the number of required
parameters of an ANN to approximate the solution of the Black-Scholes PDE grows at most polynomially in both the reciprocal of the
prescribed approximation accuracy |
|---|---|
| AbstractList | Artificial neural networks (ANNs) have very successfully been used in numerical simulations for a series of computational problems
ranging from image classification/image recognition, speech recognition, time series analysis, game intelligence, and computational
advertising to numerical approximations of partial differential equations (PDEs). Such numerical simulations suggest that ANNs have the
capacity to very efficiently approximate high-dimensional functions and, especially, indicate that ANNs seem to admit the fundamental
power to overcome the curse of dimensionality when approximating the high-dimensional functions appearing in the above named
computational problems. There are a series of rigorous mathematical approximation results for ANNs in the scientific literature. Some of
them prove convergence without convergence rates and some of these mathematical results even rigorously establish convergence rates but
there are only a few special cases where mathematical results can rigorously explain the empirical success of ANNs when approximating
high-dimensional functions. The key contribution of this article is to disclose that ANNs can efficiently approximate high-dimensional
functions in the case of numerical approximations of Black-Scholes PDEs. More precisely, this work reveals that the number of required
parameters of an ANN to approximate the solution of the Black-Scholes PDE grows at most polynomially in both the reciprocal of the
prescribed approximation accuracy View the abstract. |
| Author | Hornung, Fabian von Wurstemberger, Philippe Grohs, Philipp Jentzen, Arnulf |
| Author_xml | – sequence: 1 givenname: Philipp surname: Grohs fullname: Grohs, Philipp – sequence: 2 givenname: Fabian surname: Hornung fullname: Hornung, Fabian – sequence: 3 givenname: Arnulf surname: Jentzen fullname: Jentzen, Arnulf – sequence: 4 givenname: Philippe surname: von Wurstemberger fullname: von Wurstemberger, Philippe |
| BackLink | https://cir.nii.ac.jp/crid/1130577729740291001$$DView record in CiNii |
| BookMark | eNpVks9u1DAQxg1tEbtlD7xBJDiUQ-j4f3zcbheKVLWVQIhb5HVs1dokbuOkpbe-Q8-8HE-Ck90LF4-s-X3-ZsYzR4dtaC1C7zF8xqDgtLFNOMUMwyu0ULLATAKTjBX8NZphxWQuCMEHaD4luKDk1yGaAQieKyLEEZoTIBQwB8beJIoqoWgBBX6LFjH6DXDBKGGEztCfZXbTheCy_lb32bLrvfPG6zq7skM3hf4xdNuYXT_YzoTGJtBmq6GLNkuqc9_YNvrQ6tr3T5lvp_TV0NjOmyRf3t114bdvdJ-YUXBWa7P9-_zy3dyG2sbsRifLBJ5752xn2-myvh8mQXyHjpyuo13s4zH6-WX9Y3WRX15__bZaXuYac6z6nFaSuYprypwRVSVUURFJCgHSEc2Mw4I7R0WxUaKCDVSa80Kn4TBHDS-YoMfo0-5hHbf2Maba-lg-1HYTwjaW_31BYk92bGrtfrCxLyfMpOLTxMr12YoCpwpLSOjHHdp6Xxo_nhinrJSSKMmAKAyAE_Zh797Ecu-JoRxXoRxXoRxXgf4DdZSeIQ |
| CitedBy_id | crossref_primary_10_1214_24_AAP2071 crossref_primary_10_2139_ssrn_3782722 crossref_primary_10_1007_s10958_023_06580_1 crossref_primary_10_1142_S021953052550006X crossref_primary_10_1016_j_apenergy_2024_123874 crossref_primary_10_1109_ACCESS_2025_3529357 crossref_primary_10_1515_jnma_2024_0074 crossref_primary_10_1109_TNNLS_2021_3049719 crossref_primary_10_1016_j_jcp_2025_114273 crossref_primary_10_1111_mafi_12389 crossref_primary_10_1007_s10489_025_06669_x crossref_primary_10_1016_j_neunet_2024_106223 crossref_primary_10_1111_mafi_12443 crossref_primary_10_3934_dcdss_2025061 |
| ContentType | eBook Book |
| Copyright | Copyright 2023 American Mathematical Society |
| Copyright_xml | – notice: Copyright 2023 American Mathematical Society |
| DBID | RYH |
| DEWEY | 518/.64 |
| DOI | 10.1090/memo/1410 |
| DatabaseName | CiNii Complete |
| DatabaseTitleList | |
| DeliveryMethod | fulltext_linktorsrc |
| Discipline | Mathematics |
| EISBN | 9781470474485 1470474484 |
| EISSN | 1947-6221 |
| Edition | 1 |
| ExternalDocumentID | 9781470474485 EBC30539170 BD02181080 10_1090_memo_1410 |
| GroupedDBID | --Z -~X 123 4.4 85S ABPPZ ACNCT ACNUO AEGFZ AENEX ALMA_UNASSIGNED_HOLDINGS DU5 P2P RMA WH7 YNT YQT 38. AABBV ABARN ABQPQ ADVEM AEPJP AERYV AFOJC AHWGJ AJFER BBABE CZZ GEOUK RYH |
| ID | FETCH-LOGICAL-a1519t-3d74fd5a34fc6dd698d2728607f2a4cf165ff368b96d0b0da558a0444f3c58463 |
| ISBN | 147045632X 9781470456320 |
| ISICitedReferencesCount | 28 |
| ISICitedReferencesURI | http://www.webofscience.com/api/gateway?GWVersion=2&SrcApp=Summon&SrcAuth=ProQuest&DestLinkType=CitingArticles&DestApp=WOS_CPL&KeyUT=000991353000001&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D |
| ISSN | 0065-9266 |
| IngestDate | Mon Jun 16 09:14:45 EDT 2025 Wed Dec 10 11:21:24 EST 2025 Fri Jun 27 00:24:21 EDT 2025 Thu Aug 14 15:25:17 EDT 2025 |
| IsPeerReviewed | true |
| IsScholarly | true |
| LCCN | 2023015044 |
| LCCallNum_Ident | QA377 .G764 2023 |
| Language | English |
| LinkModel | OpenURL |
| MergedId | FETCHMERGED-LOGICAL-a1519t-3d74fd5a34fc6dd698d2728607f2a4cf165ff368b96d0b0da558a0444f3c58463 |
| Notes | April 2023, volume 284, number 1410 (sixth of 6 numbers) Other authors: Fabian Hornung, Arnulf Jentzen, Philippe von Wurstemberger Includes bibliographical references (p. 89-93) |
| OCLC | 1396938081 |
| PQID | EBC30539170 |
| PageCount | 106 |
| ParticipantIDs | askewsholts_vlebooks_9781470474485 proquest_ebookcentral_EBC30539170 nii_cinii_1130577729740291001 ams_ebooks_10_1090_memo_1410 |
| PublicationCentury | 2000 |
| PublicationDate | ♭2023. |
| PublicationDateYYYYMMDD | 2023-01-01 |
| PublicationDate_xml | – year: 2023 text: [2023] |
| PublicationDecade | 2020 |
| PublicationPlace | Providence, Rhode Island |
| PublicationPlace_xml | – name: Providence, Rhode Island – name: Providence, RI – name: Providence |
| PublicationSeriesTitle | Memoirs of the American Mathematical Society |
| PublicationYear | 2023 |
| Publisher | American Mathematical Society |
| Publisher_xml | – name: American Mathematical Society |
| SSID | ssib056432423 ssj0008047 ssib051361791 ssj0002865429 |
| Score | 2.7620068 |
| Snippet | Artificial neural networks (ANNs) have very successfully been used in numerical simulations for a series of computational problems
ranging from image... View the abstract. |
| SourceID | askewsholts proquest nii ams |
| SourceType | Aggregation Database Publisher |
| SubjectTerms | Approximation theory Differential equations, Partial-Numerical solutions Neural networks (Computer science) Stochastic differential equations |
| TableOfContents | Introduction
--
Probabilistic and analytic preliminaries
--
Artificial neural network approximations
--
Artificial neural network approximations for Black-Scholes partial differential equations Cover -- Title page -- Chapter 1. Introduction -- Acknowlegments -- Chapter 2. Probabilistic and analytic preliminaries -- 2.1. Monte Carlo approximations -- 2.2. Properties of affine functions -- 2.3. A priori estimates for solutions of stochastic differential equations -- 2.4. Stochastic differential equations with affine coefficient functions -- 2.5. Viscosity solutions for partial differential equations -- Chapter 3. Artificial neural network approximations -- 3.1. Construction of a realization on the artificial probability space -- 3.2. Approximation error estimates -- 3.3. Cost estimates -- 3.4. Representation properties for artificial neural networks -- 3.5. Cost estimates for artificial neural networks -- 3.6. Artificial neural networks with continuous activation functions -- Chapter 4. Artificial neural network approximations for Black-Scholes partial differential equations -- 4.1. Elementary properties of the Black-Scholes model -- 4.2. Transformations of viscosity solutions -- 4.3. Artificial neural network approximations for basket call options -- 4.4. Artificial neural network approximations for basket put options -- 4.5. Artificial neural network approximations for call on max options -- 4.6. Artificial neural network approximations for call on min options -- Bibliography -- Back Cover |
| Title | A Proof that Artificial Neural Networks Overcome the Curse of Dimensionality in the Numerical Approximation of Black–Scholes Partial Differential Equations |
| URI | https://www.ams.org/memo/1410/ https://cir.nii.ac.jp/crid/1130577729740291001 https://ebookcentral.proquest.com/lib/[SITE_ID]/detail.action?docID=30539170 https://www.vlebooks.com/vleweb/product/openreader?id=none&isbn=9781470474485 |
| Volume | 284 |
| WOSCitedRecordID | wos000991353000001&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D |
| hasFullText | 1 |
| inHoldings | 1 |
| isFullTextHit | |
| isPrint | |
| link | http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwtV3Pb9MwFLbawoGe-CkKGzKI2xQtaRw7Pm6jAwEaHMboLUpiZ0S0SWnaqoI_hX-W9xwnTbcD4sAlaRLbL837FH-23_tCyOtEc8wgGTtahcwBUAROmHLmJIFmksXQh5kV_KuP4uIinE7l517vV5MLs5mJogi3W7n4r66Gc-BsTJ39B3e3jcIJ-A1Ohy24HbY3GHF7aOONMdjKLPnDoB8RYcUhULLS7EzAd3WEQZtgWBvOmWIQBzJGhTL_tUQHEnMb_lis6xWdWS0-vs3nLcU0U38OqnjOdHW0QINmwaf-4Io50D_WnQlBE-VTfqt28ziLFlTlsrBvnfM46SD2PbT0s34znkCRWdZc2MBNfIV7X2n8osl1DTzbqu5OZYz9G1MZuzWqVrEWFVHKVhKlHfd6TCAV9U0e3e1ewJUYNjnX89LICdi42X1Z7dM3ht0AYe6TvuAwbr_zdvLpy4d2gg5TdqGzNsmA1tq00QhrrDdSVdI9RmvHaMtI9lZDMoyr79BPgRNWcNQv8vxWd284zOV9MsC8lgekp4uHZLj789Uj8vuEGuhQhA7dQYfW0KENdGgDHSioqYEOhVr70KF5YS630KF70MEKe9ChFjq0Cx3aQucxuTqfXJ69c-xXO5wY2KNcOb4SLFNB7LMs5UpxGaqxgKfpimwcszTzeJBlPg8TyZWbuCoOgjBG2cLMT5EO-0_IoCgL_ZTQhAUZlzrFWowrHsJoXqYwpPGkAiLqjcgBPOvIxBVUUR1P4UboighdMSKvOk6INjNbsPGhYCwMRuQQfBOlOW49oHSBwBGnYO5YokDZiLxsvFYbslHT0eT0DAr70hPus7-08Zzc2-H9gAxWy7U-JHfTzSqvli8s8v4AP8KkoQ |
| linkProvider | ProQuest Ebooks |
| openUrl | ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.title=A+proof+that+artificial+neural+networks+overcome+the+curse+of+dimensionality+in+the+numerical+approximation+of+Black-Scholes+partial+differential+equations&rft.au=Grohs%2C+Philipp&rft.au=Hornung%2C+Fabian&rft.au=Jentzen%2C+Arnulf&rft.au=von+Wurstemberger%2C+Philippe&rft.date=2023-01-01&rft.pub=American+Mathematical+Society&rft.isbn=9781470456320&rft_id=info:doi/10.1090%2Fmemo%2F1410&rft.externalDocID=BD02181080 |
| thumbnail_m | http://cvtisr.summon.serialssolutions.com/2.0.0/image/custom?url=https%3A%2F%2Fvle.dmmserver.com%2Fmedia%2F640%2F97814704%2F9781470474485.jpg |