Riemann-Stieltjes approximations of stochastic integrals View Full Text


Ontology type: schema:ScholarlyArticle     


Article Info

DATE

1969

AUTHORS

E. Wong, M. Zakai

ABSTRACT

We consider the space C[0, 1] together with its Borel σ-algebra A and a Wiener measure P. Let Ω denote a point in C[0, 1] and let x(Ω, t) denote the coordinate process. Then, {x(Ω, t), tε[0, 1]} is a Wiener process, and stochastic integrals of the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\int\limits_0^1 \varphi {\text{ }}(\omega ,t)dx(\omega ,t)$$ \end{document} can be defined for a suitable class of ϕ. In this paper we consider a sequence of Stieltjes integrals of the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$I_n = \int\limits_0^1 \varphi {\text{ }}(\omega ^n (\omega ),t)dx(\omega ^n (\omega ),t)$$ \end{document} where {Ωn(Ω)} is a sequence of polygonal approximations to co. Conditions are found which ensure the quadratic-mean convergence of {In}, and the limit is expressed as the sum of the stochastic integral \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\int\limits_0^1 \varphi {\text{ }}(\omega ,t)dx(\omega ,t)$$ \end{document} and a “correction term”. More... »

PAGES

87-97

References to SciGraph publications

Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/bf00531642

DOI

http://dx.doi.org/10.1007/bf00531642

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1007146725


Indexing Status Check whether this publication has been indexed by Scopus and Web Of Science using the SN Indexing Status Tool
Incoming Citations Browse incoming citations for this publication using opencitations.net

JSON-LD is the canonical representation for SciGraph data.

TIP: You can open this SciGraph record using an external JSON-LD service: JSON-LD Playground Google SDTT

[
  {
    "@context": "https://springernature.github.io/scigraph/jsonld/sgcontext.json", 
    "about": [
      {
        "id": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/01", 
        "inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/", 
        "name": "Mathematical Sciences", 
        "type": "DefinedTerm"
      }, 
      {
        "id": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/0104", 
        "inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/", 
        "name": "Statistics", 
        "type": "DefinedTerm"
      }
    ], 
    "author": [
      {
        "affiliation": {
          "alternateName": "College of Engineering, University of California, Berkeley, California, USA", 
          "id": "http://www.grid.ac/institutes/grid.47840.3f", 
          "name": [
            "College of Engineering, University of California, Berkeley, California, USA"
          ], 
          "type": "Organization"
        }, 
        "familyName": "Wong", 
        "givenName": "E.", 
        "id": "sg:person.012775711513.70", 
        "sameAs": [
          "https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.012775711513.70"
        ], 
        "type": "Person"
      }, 
      {
        "affiliation": {
          "alternateName": "Technion-Israel Institute of Technology, Faculty of Electrical Engineering, Haifa, Israel", 
          "id": "http://www.grid.ac/institutes/grid.6451.6", 
          "name": [
            "Technion-Israel Institute of Technology, Faculty of Electrical Engineering, Haifa, Israel"
          ], 
          "type": "Organization"
        }, 
        "familyName": "Zakai", 
        "givenName": "M.", 
        "id": "sg:person.011256533321.99", 
        "sameAs": [
          "https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.011256533321.99"
        ], 
        "type": "Person"
      }
    ], 
    "citation": [
      {
        "id": "sg:pub.10.1007/978-3-662-00031-1", 
        "sameAs": [
          "https://app.dimensions.ai/details/publication/pub.1033493461", 
          "https://doi.org/10.1007/978-3-662-00031-1"
        ], 
        "type": "CreativeWork"
      }
    ], 
    "datePublished": "1969", 
    "datePublishedReg": "1969-01-01", 
    "description": "We consider the space C[0, 1] together with its Borel \u03c3-algebra A and a Wiener measure P. Let \u03a9 denote a point in C[0, 1] and let x(\u03a9, t) denote the coordinate process. Then, {x(\u03a9, t), t\u03b5[0, 1]} is a Wiener process, and stochastic integrals of the form \\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym}\n\t\t\t\t\\usepackage{amsfonts}\n\t\t\t\t\\usepackage{amssymb}\n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}\n$$\\int\\limits_0^1 \\varphi  {\\text{ }}(\\omega ,t)dx(\\omega ,t)$$\n\\end{document} can be defined for a suitable class of \u03d5. In this paper we consider a sequence of Stieltjes integrals of the form \\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym}\n\t\t\t\t\\usepackage{amsfonts}\n\t\t\t\t\\usepackage{amssymb}\n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}\n$$I_n  = \\int\\limits_0^1 \\varphi  {\\text{ }}(\\omega ^n (\\omega ),t)dx(\\omega ^n (\\omega ),t)$$\n\\end{document} where {\u03a9n(\u03a9)} is a sequence of polygonal approximations to co. Conditions are found which ensure the quadratic-mean convergence of {In}, and the limit is expressed as the sum of the stochastic integral \\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym}\n\t\t\t\t\\usepackage{amsfonts}\n\t\t\t\t\\usepackage{amssymb}\n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}\n$$\\int\\limits_0^1 \\varphi  {\\text{ }}(\\omega ,t)dx(\\omega ,t)$$\n\\end{document} and a \u201ccorrection term\u201d.", 
    "genre": "article", 
    "id": "sg:pub.10.1007/bf00531642", 
    "isAccessibleForFree": false, 
    "isPartOf": [
      {
        "id": "sg:journal.1053886", 
        "issn": [
          "0178-8051", 
          "1432-2064"
        ], 
        "name": "Probability Theory and Related Fields", 
        "publisher": "Springer Nature", 
        "type": "Periodical"
      }, 
      {
        "issueNumber": "2", 
        "type": "PublicationIssue"
      }, 
      {
        "type": "PublicationVolume", 
        "volumeNumber": "12"
      }
    ], 
    "keywords": [
      "stochastic integrals", 
      "\u03c3-algebra A", 
      "quadratic-mean convergence", 
      "Stieltjes integral", 
      "Wiener process", 
      "correction term", 
      "suitable class", 
      "integrals", 
      "polygonal approximation", 
      "coordinate process", 
      "approximation", 
      "convergence", 
      "space", 
      "sum", 
      "class", 
      "form", 
      "terms", 
      "point", 
      "limit", 
      "sequence", 
      "process", 
      "conditions", 
      "p.", 
      "Co.", 
      "paper"
    ], 
    "name": "Riemann-Stieltjes approximations of stochastic integrals", 
    "pagination": "87-97", 
    "productId": [
      {
        "name": "dimensions_id", 
        "type": "PropertyValue", 
        "value": [
          "pub.1007146725"
        ]
      }, 
      {
        "name": "doi", 
        "type": "PropertyValue", 
        "value": [
          "10.1007/bf00531642"
        ]
      }
    ], 
    "sameAs": [
      "https://doi.org/10.1007/bf00531642", 
      "https://app.dimensions.ai/details/publication/pub.1007146725"
    ], 
    "sdDataset": "articles", 
    "sdDatePublished": "2022-11-24T21:05", 
    "sdLicense": "https://scigraph.springernature.com/explorer/license/", 
    "sdPublisher": {
      "name": "Springer Nature - SN SciGraph project", 
      "type": "Organization"
    }, 
    "sdSource": "s3://com-springernature-scigraph/baseset/20221124/entities/gbq_results/article/article_81.jsonl", 
    "type": "ScholarlyArticle", 
    "url": "https://doi.org/10.1007/bf00531642"
  }
]
 

Download the RDF metadata as:  json-ld nt turtle xml License info

HOW TO GET THIS DATA PROGRAMMATICALLY:

JSON-LD is a popular format for linked data which is fully compatible with JSON.

curl -H 'Accept: application/ld+json' 'https://scigraph.springernature.com/pub.10.1007/bf00531642'

N-Triples is a line-based linked data format ideal for batch operations.

curl -H 'Accept: application/n-triples' 'https://scigraph.springernature.com/pub.10.1007/bf00531642'

Turtle is a human-readable linked data format.

curl -H 'Accept: text/turtle' 'https://scigraph.springernature.com/pub.10.1007/bf00531642'

RDF/XML is a standard XML format for linked data.

curl -H 'Accept: application/rdf+xml' 'https://scigraph.springernature.com/pub.10.1007/bf00531642'


 

This table displays all metadata directly associated to this object as RDF triples.

96 TRIPLES      21 PREDICATES      51 URIs      42 LITERALS      6 BLANK NODES

Subject Predicate Object
1 sg:pub.10.1007/bf00531642 schema:about anzsrc-for:01
2 anzsrc-for:0104
3 schema:author Na1ad3c88ba14471797c24eecf271d70a
4 schema:citation sg:pub.10.1007/978-3-662-00031-1
5 schema:datePublished 1969
6 schema:datePublishedReg 1969-01-01
7 schema:description We consider the space C[0, 1] together with its Borel σ-algebra A and a Wiener measure P. Let Ω denote a point in C[0, 1] and let x(Ω, t) denote the coordinate process. Then, {x(Ω, t), tε[0, 1]} is a Wiener process, and stochastic integrals of the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\int\limits_0^1 \varphi {\text{ }}(\omega ,t)dx(\omega ,t)$$ \end{document} can be defined for a suitable class of ϕ. In this paper we consider a sequence of Stieltjes integrals of the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$I_n = \int\limits_0^1 \varphi {\text{ }}(\omega ^n (\omega ),t)dx(\omega ^n (\omega ),t)$$ \end{document} where {Ωn(Ω)} is a sequence of polygonal approximations to co. Conditions are found which ensure the quadratic-mean convergence of {In}, and the limit is expressed as the sum of the stochastic integral \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\int\limits_0^1 \varphi {\text{ }}(\omega ,t)dx(\omega ,t)$$ \end{document} and a “correction term”.
8 schema:genre article
9 schema:isAccessibleForFree false
10 schema:isPartOf N6f6c62f6e085477d8dd546be6370edf8
11 Na929e626c9754d519a3f4d1c9c1b5e79
12 sg:journal.1053886
13 schema:keywords Co.
14 Stieltjes integral
15 Wiener process
16 approximation
17 class
18 conditions
19 convergence
20 coordinate process
21 correction term
22 form
23 integrals
24 limit
25 p.
26 paper
27 point
28 polygonal approximation
29 process
30 quadratic-mean convergence
31 sequence
32 space
33 stochastic integrals
34 suitable class
35 sum
36 terms
37 σ-algebra A
38 schema:name Riemann-Stieltjes approximations of stochastic integrals
39 schema:pagination 87-97
40 schema:productId Nd9e48ed01fd4413cb12a9c313f918a05
41 Nee6dac085d964feb992d53879de62922
42 schema:sameAs https://app.dimensions.ai/details/publication/pub.1007146725
43 https://doi.org/10.1007/bf00531642
44 schema:sdDatePublished 2022-11-24T21:05
45 schema:sdLicense https://scigraph.springernature.com/explorer/license/
46 schema:sdPublisher Ned6f7c6555c14483a4a5ce5d6529d0d7
47 schema:url https://doi.org/10.1007/bf00531642
48 sgo:license sg:explorer/license/
49 sgo:sdDataset articles
50 rdf:type schema:ScholarlyArticle
51 N6f6c62f6e085477d8dd546be6370edf8 schema:volumeNumber 12
52 rdf:type schema:PublicationVolume
53 Na1ad3c88ba14471797c24eecf271d70a rdf:first sg:person.012775711513.70
54 rdf:rest Na2dce5813e634fd7b7e43d5ab60659e7
55 Na2dce5813e634fd7b7e43d5ab60659e7 rdf:first sg:person.011256533321.99
56 rdf:rest rdf:nil
57 Na929e626c9754d519a3f4d1c9c1b5e79 schema:issueNumber 2
58 rdf:type schema:PublicationIssue
59 Nd9e48ed01fd4413cb12a9c313f918a05 schema:name doi
60 schema:value 10.1007/bf00531642
61 rdf:type schema:PropertyValue
62 Ned6f7c6555c14483a4a5ce5d6529d0d7 schema:name Springer Nature - SN SciGraph project
63 rdf:type schema:Organization
64 Nee6dac085d964feb992d53879de62922 schema:name dimensions_id
65 schema:value pub.1007146725
66 rdf:type schema:PropertyValue
67 anzsrc-for:01 schema:inDefinedTermSet anzsrc-for:
68 schema:name Mathematical Sciences
69 rdf:type schema:DefinedTerm
70 anzsrc-for:0104 schema:inDefinedTermSet anzsrc-for:
71 schema:name Statistics
72 rdf:type schema:DefinedTerm
73 sg:journal.1053886 schema:issn 0178-8051
74 1432-2064
75 schema:name Probability Theory and Related Fields
76 schema:publisher Springer Nature
77 rdf:type schema:Periodical
78 sg:person.011256533321.99 schema:affiliation grid-institutes:grid.6451.6
79 schema:familyName Zakai
80 schema:givenName M.
81 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.011256533321.99
82 rdf:type schema:Person
83 sg:person.012775711513.70 schema:affiliation grid-institutes:grid.47840.3f
84 schema:familyName Wong
85 schema:givenName E.
86 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.012775711513.70
87 rdf:type schema:Person
88 sg:pub.10.1007/978-3-662-00031-1 schema:sameAs https://app.dimensions.ai/details/publication/pub.1033493461
89 https://doi.org/10.1007/978-3-662-00031-1
90 rdf:type schema:CreativeWork
91 grid-institutes:grid.47840.3f schema:alternateName College of Engineering, University of California, Berkeley, California, USA
92 schema:name College of Engineering, University of California, Berkeley, California, USA
93 rdf:type schema:Organization
94 grid-institutes:grid.6451.6 schema:alternateName Technion-Israel Institute of Technology, Faculty of Electrical Engineering, Haifa, Israel
95 schema:name Technion-Israel Institute of Technology, Faculty of Electrical Engineering, Haifa, Israel
96 rdf:type schema:Organization
 




Preview window. Press ESC to close (or click here)


...