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 We also give an elegant statement, inspired by Peter McCullagh's unpublished work, of this result in terms of game-theoretic probability theory.

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    File URL: http://arxiv.org/pdf/0904.4364
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    Bibliographic Info

    Paper provided by arXiv.org in its series Papers with number 0904.4364.

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    Date of creation: Apr 2009
    Date of revision: Aug 2010
    Handle: RePEc:arx:papers:0904.4364

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    1. Kei Takeuchi & Masayuki Kumon & Akimichi Takemura, 2007. "A new formulation of asset trading games in continuous time with essential forcing of variation exponent," Papers, arXiv.org 0708.0275, arXiv.org, revised Jan 2010.
    2. V. Vovk, 1993. "Forecasting point and continuous processes: Prequential analysis," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer, Springer, vol. 2(1), pages 189-217, December.
    3. Masayuki Kumon & Akimichi Takemura & Kei Takeuchi, 2005. "Capital process and optimality properties of a Bayesian Skeptic in coin-tossing games," Papers, arXiv.org math/0510662, arXiv.org, revised Sep 2008.
    4. Vladimir Vovk, 2007. "Continuous-time trading and emergence of randomness," Papers, arXiv.org 0712.1275, arXiv.org, revised Dec 2007.
    5. Vladimir Vovk, 2007. "Continuous-time trading and emergence of volatility," Papers, arXiv.org 0712.1483, arXiv.org, revised Dec 2007.
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    Cited by:
    1. Vladimir Vovk, 2010. "Rough paths in idealized financial markets," Papers, arXiv.org 1005.0279, arXiv.org, revised May 2011.

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