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

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

2007-08-02

arxiv.org/abs/0708.0275v2

Bernoulli 2009, Vol. 15, No. 4, 1243-1258

Economy

Quantitative Finance

Trading and Market Microstructure

Published in at http://dx.doi.org/10.3150/08-BEJ188 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statisti

Scientific paper

10.3150/08-BEJ188

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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