## Bernoulli

• Bernoulli
• Volume 15, Number 4 (2009), 1243-1258.

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

#### 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.

#### Article information

Source
Bernoulli, Volume 15, Number 4 (2009), 1243-1258.

Dates
First available in Project Euclid: 8 January 2010

Permanent link to this document
https://projecteuclid.org/euclid.bj/1262962234

Digital Object Identifier
doi:10.3150/08-BEJ188

Mathematical Reviews number (MathSciNet)
MR2597591

Zentralblatt MATH identifier
1201.91017

#### Citation

Takeuchi, Kei; Kumon, Masayuki; Takemura, Akimichi. A new formulation of asset trading games in continuous time with essential forcing of variation exponent. Bernoulli 15 (2009), no. 4, 1243--1258. doi:10.3150/08-BEJ188. https://projecteuclid.org/euclid.bj/1262962234

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