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Continuous-time trading and the emergence of probability

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  • Vladimir Vovk

Abstract

This paper establishes a non-stochastic analogue 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 path, 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 We also give an elegant statement, inspired by Peter McCullagh's unpublished work, of this result in terms of game-theoretic probability theory.

Suggested Citation

  • Vladimir Vovk, 2009. "Continuous-time trading and the emergence of probability," Papers 0904.4364, arXiv.org, revised May 2015.
    • Handle: RePEc:arx:papers:0904.4364
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    References listed on IDEAS

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    1. Vladimir Vovk, 2007. "Continuous-time trading and emergence of volatility," Papers 0712.1483, arXiv.org, revised Dec 2007.
    2. 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.
    3. 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.
    4. Vladimir Vovk, 2007. "Continuous-time trading and emergence of randomness," Papers 0712.1275, arXiv.org, revised Dec 2007.
    5. Kei Takeuchi & Masayuki Kumon & Akimichi Takemura, 2007. "A new formulation of asset trading games in continuous time with essential forcing of variation exponent," Papers 0708.0275, arXiv.org, revised Jan 2010.
    6. 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.
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    Citations

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    Cited by:

    1. Vladimir Vovk, 2016. "Getting rich quick with the Axiom of Choice," Papers 1604.00596, arXiv.org, revised Mar 2017.
    2. Vladimir Vovk & Glenn Shafer, 2017. "Towards a probability-free theory of continuous martingales," Papers 1703.08715, arXiv.org.
    3. Vladimir Vovk & Glenn Shafer, 2016. "A probability-free and continuous-time explanation of the equity premium and CAPM," Papers 1607.00830, arXiv.org.
    4. Vladimir Vovk, 2017. "The role of measurability in game-theoretic probability," Finance and Stochastics, Springer, vol. 21(3), pages 719-739, July.
    5. Vladimir Vovk, 2015. "Purely pathwise probability-free Ito integral," Papers 1512.01698, arXiv.org, revised Jun 2016.
    6. Vladimir Vovk, 2016. "Another example of duality between game-theoretic and measure-theoretic probability," Papers 1608.02706, arXiv.org.
    7. Vladimir Vovk, 2010. "Rough paths in idealized financial markets," Papers 1005.0279, arXiv.org, revised Nov 2016.

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