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Sequential optimizing strategy in multi-dimensional bounded forecasting games

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  • Kumon, Masayuki
  • Takemura, Akimichi
  • Takeuchi, Kei

Abstract

We propose a sequential optimizing betting strategy in the multi-dimensional bounded forecasting game in the framework of game-theoretic probability of Shafer and Vovk (2001) [10]. By studying the asymptotic behavior of its capital process, we prove a generalization of the strong law of large numbers, where the convergence rate of the sample mean vector depends on the growth rate of the quadratic variation process. The growth rate of the quadratic variation process may be slower than the number of rounds or may even be zero. We also introduce an information criterion for selecting efficient betting items. These results are then applied to multiple asset trading strategies in discrete-time and continuous-time games. In the case of a continuous-time game we present a measure of the jaggedness of a vector-valued continuous process. Our results are examined by several numerical examples.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:121:y:2011:i:1:p:155-183
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    References listed on IDEAS

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    1. Erik Ordentlich & Thomas M. Cover, 1998. "The Cost of Achieving the Best Portfolio in Hindsight," Mathematics of Operations Research, INFORMS, vol. 23(4), pages 960-982, November.
    2. Masayuki Kumon & Akimichi Takemura, 2008. "On a simple strategy weakly forcing the strong law of large numbers in the bounded forecasting game," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 60(4), pages 801-812, December.
    3. 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.
    4. Anderson, T. W. & Takemura, Akimichi, 1982. "A new proof of admissibility of tests in the multivariate analysis of variance," Journal of Multivariate Analysis, Elsevier, vol. 12(4), pages 457-468, December.
    5. Thomas M. Cover, 1991. "Universal Portfolios," Mathematical Finance, Wiley Blackwell, vol. 1(1), pages 1-29, January.
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    Cited by:

    1. Sato, Ryosuke & Miyabe, Kenshi & Takemura, Akimichi, 2018. "Relation between the rate of convergence of strong law of large numbers and the rate of concentration of Bayesian prior in game-theoretic probability," Stochastic Processes and their Applications, Elsevier, vol. 128(5), pages 1466-1484.
    2. Vladimir Vovk, 2011. "Ito calculus without probability in idealized financial markets," Papers 1108.0799, arXiv.org, revised Aug 2014.
    3. Vladimir Vovk, 2012. "Continuous-time trading and the emergence of probability," Finance and Stochastics, Springer, vol. 16(4), pages 561-609, October.

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