Skip to main content
SpringerLink
Log in

Itô Calculus without Probability in Idealized Financial Markets*

  • Published:
Lithuanian Mathematical Journal Aims and scope Submit manuscript

Abstract

We consider idealized financial markets in which price paths of the traded securities are càdlàg functions, imposing mild restrictions on the allowed size of jumps. We prove the existence of quadratic variation for typical price paths, where the qualification “typical” means that there is a trading strategy that risks only one monetary unit and brings infinite capital if quadratic variation does not exist. This result allows one to apply numerous known results in pathwise Itô calculus to typical price paths; we give a brief overview of such results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. P. Billingsley, Convergence of Probability Measures, Wiley, New York, 1968.

    MATH  Google Scholar 

  2. M. Bruneau, Sur la p-variation d’une surmartingale continue, in Séminaire de Probabilités XIII, Université de Strasbourg 1977/1978, Lect. Notes Math., Vol. 721, Springer, Berlin, 1979, pp. 227–232.

  3. R. Cont and D.-A. Fournié, Change of variable formulas for non-anticipative functionals on path space, J. Funct. Anal., 259:1043–1072, 2010.

    Article  MATH  MathSciNet  Google Scholar 

  4. C. Dellacherie and P.-A. Meyer, Probabilities and Potential, North-Holland, Amsterdam, 1978.

    MATH  Google Scholar 

  5. Federal Reserve Board, Credit by brokers and dealers (Regulation T), 12 CFR 220.12, 2014.

  6. H. Föllmer, Calcul d’Itô sans probabilités, in Séminaire de Probabilités XV, Université de Strasbourg 1979/1980, Lect. Notes Math., Vol. 850, Springer, Berlin, 1981, pp. 143–150.

  7. H. Föllmer and A. Schied, Stochastic Finance: An Introduction in Discrete Time, 3rd ed., De Gruyter, Berlin, 2011.

    Book  Google Scholar 

  8. P. Fortune, Margin requirements, margin loans, and margin rates: Practice and principles, New Engl. Econ. Rev., September/October, pp. 19–44, 2000.

  9. M. Kumon K. Takeuchi and A. Takemura, A new formulation of asset trading games in continuous time with essential forcing of variation exponent, Bernoulli, 15:1243–1258, 2009.

    Article  MATH  MathSciNet  Google Scholar 

  10. A.N. Kolmogorov, Sur la loi des grands nombres, Atti Accad. Naz. Lincei, Rend., VI. Ser., 185:917–919, 1929.

  11. P.-A. Meyer, Un cours sur les intégrales stochastiques, in Séminaire de Probabilités X, Université de Strasbourg, Lect. Notes Math., Vol. 511, Springer, Berlin, 1976, pp. 245–400.

  12. K. Miyabe and A. Takemura, Convergence of random series and the rate of convergence of the strong law of large numbers in game-theoretic probability, Stochastic Processes Appl., 122:1–30, 2012.

    Article  MATH  MathSciNet  Google Scholar 

  13. K. Miyabe and A. Takemura, The law of the iterated logarithm in game-theoretic probability with quadratic and stronger hedges, Stochastic Processes Appl., 123:3132–3152, 2013.

    Article  MATH  MathSciNet  Google Scholar 

  14. K. Miyabe and A. Takemura, Derandomization in game-theoretic probability, Stochastic Processes Appl., 125:39–59, 2015.

    Article  MATH  MathSciNet  Google Scholar 

  15. R. Norvaiša, Quadratic variation, p-variation and integration with applications to stock price modelling, Technical Report, 2001, arXiv:0108090.

  16. N. Perkowski and D.J. Prömel, Pathwise stochastic integrals for model free finance, Technical Report, 2013, arXiv:1311.6187.

  17. N. Perkowski and D.J. Prömel, Local times for typical price paths and pathwise Tanaka formulas, Technical Report, 2014, arXiv:1405.4421.

  18. G. Shafer and V. Vovk, Probability and Finance: It’s Only a Game!, Wiley, New York, 2001.

    Book  Google Scholar 

  19. C. Stricker, Sur la p-variation des surmartingales, in Séminaire de Probabilités XIII, Université de Strasbourg, 1977/1978, Lect. Notes Math., Vol. 721, Springer, Berlin, 1979, pp. 233–237.

  20. V. Vovk, A logic of probability, with application to the foundations of statistics (with discussion), J. R. Stat. Soc., Ser. B, 55:317–351, 1993.

    MATH  MathSciNet  Google Scholar 

  21. V. Vovk, Rough paths in idealized financial markets, Lith. Math. J., 51:274–285, 2011. (Expanded and corrected version: arXiv:1005.0279.)

  22. V. Vovk, Continuous-time trading and the emergence of probability, Finance Stoch., 16:561–609, 2012.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, Royal Holloway, University of London, Egham, Surrey, TW20 0EX, United Kingdom

    Vladimir Vovk

Authors
  1. Vladimir Vovk
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vladimir Vovk.

Additional information

*This research was supported by the Air Force Office of Scientific Research (grant FA9550-14-1-0043).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Vovk, V. Itô Calculus without Probability in Idealized Financial Markets* . Lith Math J 55, 270–290 (2015). https://doi.org/10.1007/s10986-015-9280-1

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10986-015-9280-1

MSC

Keywords

Search

Navigation