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Continuous-time trading and the emergence of probability

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  • Vladimir Vovk

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    Abstract

    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 We also give an elegant statement, inspired by Peter McCullagh’s unpublished work, of this result in terms of game-theoretic probability theory.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

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    File URL: http://hdl.handle.net/10.1007/s00780-012-0180-5
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    Bibliographic Info

    Article provided by Springer in its journal Finance and Stochastics.

    Volume (Year): 16 (2012)
    Issue (Month): 4 (October)
    Pages: 561-609

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    Handle: RePEc:spr:finsto:v:16:y:2012:i:4:p:561-609

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    Related research

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

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    References

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    1. 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.
    2. 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.
    3. Vladimir Vovk, 2010. "Rough paths in idealized financial markets," Papers 1005.0279, arXiv.org, revised May 2011.
    4. 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.
    5. Masayuki Kumon & Akimichi Takemura & Kei Takeuchi, 2005. "Capital process and optimality properties of a Bayesian Skeptic in coin-tossing games," Papers math/0510662, arXiv.org, revised Sep 2008.
    6. David G. Hobson, 1998. "Robust hedging of the lookback option," Finance and Stochastics, Springer, vol. 2(4), pages 329-347.
    7. Gianluca Cassese, 2008. "Asset Pricing With No Exogenous Probability Measure," Mathematical Finance, Wiley Blackwell, vol. 18(1), pages 23-54.
    8. V. Vovk, 1993. "Forecasting point and continuous processes: Prequential analysis," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer, vol. 2(1), pages 189-217, December.
    9. Bick, Avi & Willinger, Walter, 1994. "Dynamic spanning without probabilities," Stochastic Processes and their Applications, Elsevier, vol. 50(2), pages 349-374, April.
    10. 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.
    11. Mathias Beiglb\"ock & Walter Schachermayer & Bezirgen Veliyev, 2010. "A Direct Proof of the Bichteler--Dellacherie Theorem and Connections to Arbitrage," Papers 1004.5559, arXiv.org.
    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.
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