Continuity of the Itô-Map for Holder Rough Paths with Applications to the Support Theorem in Holder Norm View Full Text


Ontology type: schema:Chapter     


Chapter Info

DATE

2005

AUTHORS

Peter K. Friz

ABSTRACT

Rough Path theory is currently formulated in p-variation topology. We show that in the context of Brownian motion, enhanced to a Rough Path, a more natural Holder metric π can be used. Based on fine-estimates in Lyons’ celebrated Universal Limit Theorem we obtain Lipschitz-continuity of the Ito-rnap (between Rough Path spaces equipped with π). We then consider a number of approximations to Brownian Rough Paths and establish their convergence w.r.t. π. In combination with our Holder ULT this allows sharper results than the p-variation theory. Also, our formulation avoids the so-called control functions and may be easier to use for non Rough Path specialists. As concrete application, we combine our results with ideas from [MS] and [LQZ] and obtain the Stroock-Varadhan Support Theorem in Holder topology as immediate corollary. More... »

PAGES

117-135

Book

TITLE

Probability and Partial Differential Equations in Modern Applied Mathematics

ISBN

978-0-387-25879-9
978-0-387-29371-4

Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/978-0-387-29371-4_8

DOI

http://dx.doi.org/10.1007/978-0-387-29371-4_8

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1022676715


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/2008/0101", 
        "inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/", 
        "name": "Pure Mathematics", 
        "type": "DefinedTerm"
      }
    ], 
    "author": [
      {
        "affiliation": {
          "alternateName": "Courant Institute, NYU, 10012, New York, NY, USA", 
          "id": "http://www.grid.ac/institutes/grid.482020.c", 
          "name": [
            "Courant Institute, NYU, 10012, New York, NY, USA"
          ], 
          "type": "Organization"
        }, 
        "familyName": "Friz", 
        "givenName": "Peter K.", 
        "id": "sg:person.010663776615.90", 
        "sameAs": [
          "https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.010663776615.90"
        ], 
        "type": "Person"
      }
    ], 
    "datePublished": "2005", 
    "datePublishedReg": "2005-01-01", 
    "description": "Rough Path theory is currently formulated in p-variation topology. We show that in the context of Brownian motion, enhanced to a Rough Path, a more natural Holder metric \u03c0 can be used. Based on fine-estimates in Lyons\u2019 celebrated Universal Limit Theorem we obtain Lipschitz-continuity of the Ito-rnap (between Rough Path spaces equipped with \u03c0). We then consider a number of approximations to Brownian Rough Paths and establish their convergence w.r.t. \u03c0. In combination with our Holder ULT this allows sharper results than the p-variation theory. Also, our formulation avoids the so-called control functions and may be easier to use for non Rough Path specialists. As concrete application, we combine our results with ideas from [MS] and [LQZ] and obtain the Stroock-Varadhan Support Theorem in Holder topology as immediate corollary.", 
    "editor": [
      {
        "familyName": "Waymire", 
        "givenName": "Edward C.", 
        "type": "Person"
      }, 
      {
        "familyName": "Duan", 
        "givenName": "Jinqiao", 
        "type": "Person"
      }
    ], 
    "genre": "chapter", 
    "id": "sg:pub.10.1007/978-0-387-29371-4_8", 
    "isAccessibleForFree": false, 
    "isPartOf": {
      "isbn": [
        "978-0-387-25879-9", 
        "978-0-387-29371-4"
      ], 
      "name": "Probability and Partial Differential Equations in Modern Applied Mathematics", 
      "type": "Book"
    }, 
    "keywords": [
      "rough paths", 
      "support theorem", 
      "Stroock\u2013Varadhan support theorem", 
      "Universal Limit Theorem", 
      "rough path theory", 
      "Brownian rough path", 
      "number of approximations", 
      "It\u00f4 map", 
      "convergence w.", 
      "limit theorem", 
      "Lipschitz continuity", 
      "Brownian motion", 
      "sharp results", 
      "Holder norms", 
      "path theory", 
      "immediate corollary", 
      "theorem", 
      "control functions", 
      "concrete applications", 
      "topology", 
      "theory", 
      "approximation", 
      "path", 
      "corollary", 
      "motion", 
      "formulation", 
      "applications", 
      "norms", 
      "results", 
      "function", 
      "continuity", 
      "idea", 
      "number", 
      "Lyon", 
      "combination", 
      "context", 
      "holder", 
      "ULT", 
      "specialists"
    ], 
    "name": "Continuity of the It\u00f4-Map for Holder Rough Paths with Applications to the Support Theorem in Holder Norm", 
    "pagination": "117-135", 
    "productId": [
      {
        "name": "dimensions_id", 
        "type": "PropertyValue", 
        "value": [
          "pub.1022676715"
        ]
      }, 
      {
        "name": "doi", 
        "type": "PropertyValue", 
        "value": [
          "10.1007/978-0-387-29371-4_8"
        ]
      }
    ], 
    "publisher": {
      "name": "Springer Nature", 
      "type": "Organisation"
    }, 
    "sameAs": [
      "https://doi.org/10.1007/978-0-387-29371-4_8", 
      "https://app.dimensions.ai/details/publication/pub.1022676715"
    ], 
    "sdDataset": "chapters", 
    "sdDatePublished": "2022-10-01T06:56", 
    "sdLicense": "https://scigraph.springernature.com/explorer/license/", 
    "sdPublisher": {
      "name": "Springer Nature - SN SciGraph project", 
      "type": "Organization"
    }, 
    "sdSource": "s3://com-springernature-scigraph/baseset/20221001/entities/gbq_results/chapter/chapter_280.jsonl", 
    "type": "Chapter", 
    "url": "https://doi.org/10.1007/978-0-387-29371-4_8"
  }
]
 

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.10.1007/978-0-387-29371-4_8'

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

curl -H 'Accept: application/n-triples' 'https://scigraph.springernature.com/pub.10.1007/978-0-387-29371-4_8'

Turtle is a human-readable linked data format.

curl -H 'Accept: text/turtle' 'https://scigraph.springernature.com/pub.10.1007/978-0-387-29371-4_8'

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

curl -H 'Accept: application/rdf+xml' 'https://scigraph.springernature.com/pub.10.1007/978-0-387-29371-4_8'


 

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

103 TRIPLES      22 PREDICATES      64 URIs      57 LITERALS      7 BLANK NODES

Subject Predicate Object
1 sg:pub.10.1007/978-0-387-29371-4_8 schema:about anzsrc-for:01
2 anzsrc-for:0101
3 schema:author N1c63adae85b24ab5a4c3136be7d84fd8
4 schema:datePublished 2005
5 schema:datePublishedReg 2005-01-01
6 schema:description Rough Path theory is currently formulated in p-variation topology. We show that in the context of Brownian motion, enhanced to a Rough Path, a more natural Holder metric π can be used. Based on fine-estimates in Lyons’ celebrated Universal Limit Theorem we obtain Lipschitz-continuity of the Ito-rnap (between Rough Path spaces equipped with π). We then consider a number of approximations to Brownian Rough Paths and establish their convergence w.r.t. π. In combination with our Holder ULT this allows sharper results than the p-variation theory. Also, our formulation avoids the so-called control functions and may be easier to use for non Rough Path specialists. As concrete application, we combine our results with ideas from [MS] and [LQZ] and obtain the Stroock-Varadhan Support Theorem in Holder topology as immediate corollary.
7 schema:editor N70b74e7fb7dc48608e8850dfa1b47018
8 schema:genre chapter
9 schema:isAccessibleForFree false
10 schema:isPartOf Nda3556140d8544b8a6ec3101d494e9d2
11 schema:keywords Brownian motion
12 Brownian rough path
13 Holder norms
14 Itô map
15 Lipschitz continuity
16 Lyon
17 Stroock–Varadhan support theorem
18 ULT
19 Universal Limit Theorem
20 applications
21 approximation
22 combination
23 concrete applications
24 context
25 continuity
26 control functions
27 convergence w.
28 corollary
29 formulation
30 function
31 holder
32 idea
33 immediate corollary
34 limit theorem
35 motion
36 norms
37 number
38 number of approximations
39 path
40 path theory
41 results
42 rough path theory
43 rough paths
44 sharp results
45 specialists
46 support theorem
47 theorem
48 theory
49 topology
50 schema:name Continuity of the Itô-Map for Holder Rough Paths with Applications to the Support Theorem in Holder Norm
51 schema:pagination 117-135
52 schema:productId N7a85e360c55c4999acd29205f6d4fc76
53 Nfd4981f051774ae18ca7bd4a545b5fde
54 schema:publisher N5e8c5c7d1ed84c96ac9e7609c1adf484
55 schema:sameAs https://app.dimensions.ai/details/publication/pub.1022676715
56 https://doi.org/10.1007/978-0-387-29371-4_8
57 schema:sdDatePublished 2022-10-01T06:56
58 schema:sdLicense https://scigraph.springernature.com/explorer/license/
59 schema:sdPublisher N139870c1bc9a49f4a24fe69142b3d06a
60 schema:url https://doi.org/10.1007/978-0-387-29371-4_8
61 sgo:license sg:explorer/license/
62 sgo:sdDataset chapters
63 rdf:type schema:Chapter
64 N139870c1bc9a49f4a24fe69142b3d06a schema:name Springer Nature - SN SciGraph project
65 rdf:type schema:Organization
66 N1c63adae85b24ab5a4c3136be7d84fd8 rdf:first sg:person.010663776615.90
67 rdf:rest rdf:nil
68 N31ea9d45ded84402bab2d029a1730d10 rdf:first N9816fe9a490242279f5564db958c4ac3
69 rdf:rest rdf:nil
70 N5e8c5c7d1ed84c96ac9e7609c1adf484 schema:name Springer Nature
71 rdf:type schema:Organisation
72 N70b74e7fb7dc48608e8850dfa1b47018 rdf:first N87e68cc0bb6b4947a69ca405cbec1156
73 rdf:rest N31ea9d45ded84402bab2d029a1730d10
74 N7a85e360c55c4999acd29205f6d4fc76 schema:name doi
75 schema:value 10.1007/978-0-387-29371-4_8
76 rdf:type schema:PropertyValue
77 N87e68cc0bb6b4947a69ca405cbec1156 schema:familyName Waymire
78 schema:givenName Edward C.
79 rdf:type schema:Person
80 N9816fe9a490242279f5564db958c4ac3 schema:familyName Duan
81 schema:givenName Jinqiao
82 rdf:type schema:Person
83 Nda3556140d8544b8a6ec3101d494e9d2 schema:isbn 978-0-387-25879-9
84 978-0-387-29371-4
85 schema:name Probability and Partial Differential Equations in Modern Applied Mathematics
86 rdf:type schema:Book
87 Nfd4981f051774ae18ca7bd4a545b5fde schema:name dimensions_id
88 schema:value pub.1022676715
89 rdf:type schema:PropertyValue
90 anzsrc-for:01 schema:inDefinedTermSet anzsrc-for:
91 schema:name Mathematical Sciences
92 rdf:type schema:DefinedTerm
93 anzsrc-for:0101 schema:inDefinedTermSet anzsrc-for:
94 schema:name Pure Mathematics
95 rdf:type schema:DefinedTerm
96 sg:person.010663776615.90 schema:affiliation grid-institutes:grid.482020.c
97 schema:familyName Friz
98 schema:givenName Peter K.
99 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.010663776615.90
100 rdf:type schema:Person
101 grid-institutes:grid.482020.c schema:alternateName Courant Institute, NYU, 10012, New York, NY, USA
102 schema:name Courant Institute, NYU, 10012, New York, NY, USA
103 rdf:type schema:Organization
 




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


...