IDEAS home Printed from
   My bibliography  Save this article

Continuous-time trading and the emergence of probability


  • Vladimir Vovk



This paper establishes a non-stochastic analog of the celebrated result by Dubins and Schwarz about reduction of continuous martingales to Brownian motion via time change. We consider an idealized financial security with continuous price paths, without making any stochastic assumptions. It is shown that typical price paths possess quadratic variation, where “typical” is understood in the following game-theoretic sense: there exists a trading strategy that earns infinite capital without risking more than one monetary unit if the process of quadratic variation does not exist. Replacing time by the quadratic variation process, we show that the price path becomes Brownian motion. This is essentially the same conclusion as in the Dubins–Schwarz result, except that the probabilities (constituting the Wiener measure) emerge instead of being postulated. We also give an elegant statement, inspired by Peter McCullagh’s unpublished work, of this result in terms of game-theoretic probability theory. Copyright Springer-Verlag 2012

Suggested Citation

  • Vladimir Vovk, 2012. "Continuous-time trading and the emergence of probability," Finance and Stochastics, Springer, vol. 16(4), pages 561-609, October.
  • Handle: RePEc:spr:finsto:v:16:y:2012:i:4:p:561-609
    DOI: 10.1007/s00780-012-0180-5

    Download full text from publisher

    File URL:
    Download Restriction: Access to full text is restricted to subscribers.

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    1. Gianluca Cassese, 2008. "Asset Pricing With No Exogenous Probability Measure," Mathematical Finance, Wiley Blackwell, vol. 18(1), pages 23-54, January.
    2. Masayuki Kumon & Akimichi Takemura & Kei Takeuchi, 2005. "Capital process and optimality properties of a Bayesian Skeptic in coin-tossing games," Papers math/0510662,, revised Sep 2008.
    3. Bick, Avi & Willinger, Walter, 1994. "Dynamic spanning without probabilities," Stochastic Processes and their Applications, Elsevier, vol. 50(2), pages 349-374, April.
    4. Dawid, A. Philip & de Rooij, Steven & Shafer, Glenn & Shen, Alexander & Vereshchagin, Nikolai & Vovk, Vladimir, 2011. "Insuring against loss of evidence in game-theoretic probability," Statistics & Probability Letters, Elsevier, vol. 81(1), pages 157-162, January.
    5. David G. Hobson, 1998. "Robust hedging of the lookback option," Finance and Stochastics, Springer, vol. 2(4), pages 329-347.
    6. Vladimir Vovk, 2010. "Rough paths in idealized financial markets," Papers 1005.0279,, revised Nov 2016.
    7. Horikoshi, Yasunori & Takemura, Akimichi, 2008. "Implications of contrarian and one-sided strategies for the fair-coin game," Stochastic Processes and their Applications, Elsevier, vol. 118(11), pages 2125-2142, November.
    8. Alexander Cox & Jan Obłój, 2011. "Robust pricing and hedging of double no-touch options," Finance and Stochastics, Springer, vol. 15(3), pages 573-605, September.
    9. V. Vovk, 1993. "Forecasting point and continuous processes: Prequential analysis," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 2(1), pages 189-217, December.
    10. Mathias Beiglbock & Walter Schachermayer & Bezirgen Veliyev, 2010. "A Direct Proof of the Bichteler--Dellacherie Theorem and Connections to Arbitrage," Papers 1004.5559,
    11. T. J. Lyons, 1995. "Uncertain volatility and the risk-free synthesis of derivatives," Applied Mathematical Finance, Taylor & Francis Journals, vol. 2(2), pages 117-133.
    12. Kumon, Masayuki & Takemura, Akimichi & Takeuchi, Kei, 2011. "Sequential optimizing strategy in multi-dimensional bounded forecasting games," Stochastic Processes and their Applications, Elsevier, vol. 121(1), pages 155-183, January.
    Full references (including those not matched with items on IDEAS)


    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.

    Cited by:

    1. repec:eee:spapps:v:128:y:2018:i:12:p:4078-4103 is not listed on IDEAS
    2. Nicolas Perkowski & David J. Promel, 2013. "Pathwise stochastic integrals for model free finance," Papers 1311.6187,, revised Jun 2016.
    3. Lesiba Ch. Galane & Rafa{l} M. {L}ochowski & Farai J. Mhlanga, 2017. "On the quadratic variation of the model-free price paths with jumps," Papers 1710.07894,, revised May 2018.
    4. repec:spr:finsto:v:22:y:2018:i:3:d:10.1007_s00780-018-0363-9 is not listed on IDEAS
    5. Matteo Burzoni & Marco Maggis, 2019. "Arbitrage-free modeling under Knightian Uncertainty," Papers 1909.04602,
    6. Gianluca Cassese, 2017. "Asset pricing in an imperfect world," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 64(3), pages 539-570, October.
    7. repec:spr:finsto:v:21:y:2017:i:4:d:10.1007_s00780-017-0338-2 is not listed on IDEAS
    8. Mathias Beiglbock & Alexander M. G. Cox & Martin Huesmann & Nicolas Perkowski & David J. Promel, 2015. "Pathwise super-replication via Vovk's outer measure," Papers 1504.03644,, revised Jul 2016.
    9. Patrick Cheridito & Matti Kiiski & David J. Promel & H. Mete Soner, 2019. "Martingale Optimal Transport Duality," Papers 1904.04644,
    10. Daniel Bartl & Michael Kupper & Ariel Neufeld, 2017. "Pathwise superhedging on prediction sets," Papers 1711.02764,, revised Oct 2019.
    11. Daniel Bartl & Michael Kupper & David J. Promel & Ludovic Tangpi, 2017. "Duality for pathwise superhedging in continuous time," Papers 1705.02933,, revised Apr 2019.
    12. Rafa{l} M. {L}ochowski, 2015. "Integration with respect to model-free price paths with jumps," Papers 1511.08194,, revised Sep 2016.
    13. repec:spr:finsto:v:21:y:2017:i:3:d:10.1007_s00780-017-0336-4 is not listed on IDEAS
    14. Lesiba Ch. Galane & Rafa{l} M. {L}ochowski & Farai J. Mhlanga, 2018. "On SDEs with Lipschitz coefficients, driven by continuous, model-free price paths," Papers 1807.05692,, revised Mar 2019.

    More about this item


    Game-theoretic probability; Continuous time; Emergence of probability; Continuous price paths; Incomplete markets; 91G99; 60G17; 60G05; 60G44; C58; G13; G14;

    JEL classification:

    • C58 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Financial Econometrics
    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing
    • G14 - Financial Economics - - General Financial Markets - - - Information and Market Efficiency; Event Studies; Insider Trading


    Access and download statistics


    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:finsto:v:16:y:2012:i:4:p:561-609. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Sonal Shukla) or (Mallaigh Nolan). General contact details of provider: .

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.