An information geometry problem in mathematical finance

I. Csiszár, Thomas Breuer

Research output: Chapter in Book/Report/Conference proceedingConference contribution

3 Citations (Scopus)

Abstract

Familiar approaches to risk and preferences involve minimizing the expectation EIP(X) of a payoff function X over a family Γ of plausible risk factor distributions IP. We consider Γ determined by a bound on a convex integral functional of the density of IP, thus Γ may be an I-divergence (relative entropy) ball or some other f-divergence ball or Bregman distance ball around a default distribution IP0. Using a Pythagorean identity we show that whether or not a worst case distribution exists (minimizing EIP(X) subject to IP ∈ Γ), the almost worst case distributions cluster around an explicitly specified, perhaps incomplete distribution. When Γ is an f-divergence ball, a worst case distribution either exists for any radius, or it does/does not exist for radius less/larger than a critical value. It remains open how far the latter result extends beyond f-divergence balls.

Original languageEnglish
Title of host publicationLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
PublisherSpringer Verlag
Pages435-443
Number of pages9
Volume9389
ISBN (Print)9783319250397, 9783319250397
DOIs
Publication statusPublished - 2015
Event2nd International Conference on Geometric Science of Information, GSI 2015 - Palaiseau, France
Duration: Oct 28 2015Oct 30 2015

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume9389
ISSN (Print)03029743
ISSN (Electronic)16113349

Other

Other2nd International Conference on Geometric Science of Information, GSI 2015
CountryFrance
CityPalaiseau
Period10/28/1510/30/15

Fingerprint

Information Geometry
Mathematical Finance
Finance
F-divergence
Ball
Geometry
Entropy
Pythagorean identity
Radius
Bregman Distance
Functional Integral
Relative Entropy
Risk Factors
Critical value
Divergence

Keywords

  • Almost worst case densities
  • Bregman distance
  • Convex integral functional
  • Fdivergence
  • I-divergence
  • Payoff function
  • Pythagorean identity
  • Risk measure
  • Worst case density

ASJC Scopus subject areas

  • Computer Science(all)
  • Theoretical Computer Science

Cite this

Csiszár, I., & Breuer, T. (2015). An information geometry problem in mathematical finance. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 9389, pp. 435-443). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 9389). Springer Verlag. https://doi.org/10.1007/978-3-319-25040-3_47

An information geometry problem in mathematical finance. / Csiszár, I.; Breuer, Thomas.

Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). Vol. 9389 Springer Verlag, 2015. p. 435-443 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 9389).

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Csiszár, I & Breuer, T 2015, An information geometry problem in mathematical finance. in Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). vol. 9389, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 9389, Springer Verlag, pp. 435-443, 2nd International Conference on Geometric Science of Information, GSI 2015, Palaiseau, France, 10/28/15. https://doi.org/10.1007/978-3-319-25040-3_47
Csiszár I, Breuer T. An information geometry problem in mathematical finance. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). Vol. 9389. Springer Verlag. 2015. p. 435-443. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). https://doi.org/10.1007/978-3-319-25040-3_47
Csiszár, I. ; Breuer, Thomas. / An information geometry problem in mathematical finance. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). Vol. 9389 Springer Verlag, 2015. pp. 435-443 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).
@inproceedings{06b2ea60457742fbb19a191d0e56699d,
title = "An information geometry problem in mathematical finance",
abstract = "Familiar approaches to risk and preferences involve minimizing the expectation EIP(X) of a payoff function X over a family Γ of plausible risk factor distributions IP. We consider Γ determined by a bound on a convex integral functional of the density of IP, thus Γ may be an I-divergence (relative entropy) ball or some other f-divergence ball or Bregman distance ball around a default distribution IP0. Using a Pythagorean identity we show that whether or not a worst case distribution exists (minimizing EIP(X) subject to IP ∈ Γ), the almost worst case distributions cluster around an explicitly specified, perhaps incomplete distribution. When Γ is an f-divergence ball, a worst case distribution either exists for any radius, or it does/does not exist for radius less/larger than a critical value. It remains open how far the latter result extends beyond f-divergence balls.",
keywords = "Almost worst case densities, Bregman distance, Convex integral functional, Fdivergence, I-divergence, Payoff function, Pythagorean identity, Risk measure, Worst case density",
author = "I. Csisz{\'a}r and Thomas Breuer",
year = "2015",
doi = "10.1007/978-3-319-25040-3_47",
language = "English",
isbn = "9783319250397",
volume = "9389",
series = "Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)",
publisher = "Springer Verlag",
pages = "435--443",
booktitle = "Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)",

}

TY - GEN

T1 - An information geometry problem in mathematical finance

AU - Csiszár, I.

AU - Breuer, Thomas

PY - 2015

Y1 - 2015

N2 - Familiar approaches to risk and preferences involve minimizing the expectation EIP(X) of a payoff function X over a family Γ of plausible risk factor distributions IP. We consider Γ determined by a bound on a convex integral functional of the density of IP, thus Γ may be an I-divergence (relative entropy) ball or some other f-divergence ball or Bregman distance ball around a default distribution IP0. Using a Pythagorean identity we show that whether or not a worst case distribution exists (minimizing EIP(X) subject to IP ∈ Γ), the almost worst case distributions cluster around an explicitly specified, perhaps incomplete distribution. When Γ is an f-divergence ball, a worst case distribution either exists for any radius, or it does/does not exist for radius less/larger than a critical value. It remains open how far the latter result extends beyond f-divergence balls.

AB - Familiar approaches to risk and preferences involve minimizing the expectation EIP(X) of a payoff function X over a family Γ of plausible risk factor distributions IP. We consider Γ determined by a bound on a convex integral functional of the density of IP, thus Γ may be an I-divergence (relative entropy) ball or some other f-divergence ball or Bregman distance ball around a default distribution IP0. Using a Pythagorean identity we show that whether or not a worst case distribution exists (minimizing EIP(X) subject to IP ∈ Γ), the almost worst case distributions cluster around an explicitly specified, perhaps incomplete distribution. When Γ is an f-divergence ball, a worst case distribution either exists for any radius, or it does/does not exist for radius less/larger than a critical value. It remains open how far the latter result extends beyond f-divergence balls.

KW - Almost worst case densities

KW - Bregman distance

KW - Convex integral functional

KW - Fdivergence

KW - I-divergence

KW - Payoff function

KW - Pythagorean identity

KW - Risk measure

KW - Worst case density

UR - http://www.scopus.com/inward/record.url?scp=84950327509&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84950327509&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-25040-3_47

DO - 10.1007/978-3-319-25040-3_47

M3 - Conference contribution

SN - 9783319250397

SN - 9783319250397

VL - 9389

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 435

EP - 443

BT - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

PB - Springer Verlag

ER -