No. 1042

Title:

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

Author(s):

Kei, Takeuchi (Faculty of International Studies, Meiji Gakuin University);
Masayuki, Kumon (Risk Analysis Research Center, Institute of Statistical Mathematics);
Akimichi, Takemura (Graduate School of Information Science and Technology, University of Tokyo)

Key words:

Baysian strategy, beta-binomial distribution, game-theorethic probability, Holder exponent, Kullback-Leibler information, modulus of continuity, square root of dt effect.

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 [13]. 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 [10] for discrete time coin-tossing games. We also clarify that the main growth part of the investor's capital processes is lucidly described by the information quantities, which are derived from the Kullback-Leibler information with respect to the empirical fluctuation of the asset price.