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A new formulation of asset trading games in continuous time with essential forcing of variation exponent

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  • Kei Takeuchi
  • Masayuki Kumon
  • Akimichi Takemura
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    Abstract

    We introduce a new formulation of asset trading games in continuous time in the framework of the game-theoretic probability established by Shafer and Vovk (Probability and Finance: It's Only a Game! (2001) Wiley). In our formulation, the market moves continuously, but an investor trades in discrete times, which can depend on the past path of the market. We prove that an investor can essentially force that the asset price path behaves with the variation exponent exactly equal to two. Our proof is based on embedding high-frequency discrete-time games into the continuous-time game and the use of the Bayesian strategy of Kumon, Takemura and Takeuchi (Stoch. Anal. Appl. 26 (2008) 1161--1180) for discrete-time coin-tossing games. We also show that the main growth part of the investor's capital processes is clearly described by the information quantities, which are derived from the Kullback--Leibler information with respect to the empirical fluctuation of the asset price.

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    File URL: http://arxiv.org/pdf/0708.0275
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    Bibliographic Info

    Paper provided by arXiv.org in its series Papers with number 0708.0275.

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    Date of creation: Aug 2007
    Date of revision: Jan 2010
    Publication status: Published in Bernoulli 2009, Vol. 15, No. 4, 1243-1258
    Handle: RePEc:arx:papers:0708.0275

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    Web page: http://arxiv.org/

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    References

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    1. 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.
    2. Yasunori Horikoshi & Akimichi Takemura, 2007. "Implications of contrarian and one-sided strategies for the fair-coin game," Papers math/0703743, arXiv.org.
    3. Thomas M. Cover, 1991. "Universal Portfolios," Mathematical Finance, Wiley Blackwell, vol. 1(1), pages 1-29.
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    Cited by:
    1. Vladimir Vovk, 2009. "Continuous-time trading and the emergence of probability," Papers 0904.4364, arXiv.org, revised Aug 2010.
    2. Vladimir Vovk, 2007. "Continuous-time trading and emergence of randomness," Papers 0712.1275, arXiv.org, revised Dec 2007.

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