[Paper Review] A new formulation of asset trading games in continuous time with essential forcing of variation exponent

Kei Takeuchi, Masayuki Kumon+1 more|RePEc: Research Papers in Economics|Aug 2, 2007

Financial Markets and Investment Strategies4 citations

TL;DR

This paper introduces a new game-theoretic framework for continuous-time asset trading games where an investor trades at discrete, path-dependent times, enabling the investor to essentially force the variation exponent of the asset price path to be exactly two. By embedding high-frequency discrete-time Bayesian strategies into continuous time and leveraging Kullback–Leibler information, the investor can achieve exponential capital growth if the market path deviates from quadratic variation, even without assuming stochastic processes like fractional Brownian motion.

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.

Motivation & Objective

• Develop a tractable continuous-time formulation of asset trading games within game-theoretic probability, avoiding nonstandard analysis.
• Enable discrete-time trading strategies with path-dependent timing to influence the long-term behavior of continuous price paths.
• Establish conditions under which an investor can force the variation exponent of the market path to be exactly two, regardless of the market's actual behavior.
• Characterize the growth of the investor's capital in terms of Kullback–Leibler divergence from empirical fluctuations, linking information theory to financial strategy.
• Provide pathwise results that hold for individual continuous price paths, avoiding probabilistic assumptions on market dynamics.

Proposed method

• Formulate a continuous-time game where the market moves continuously, but the investor trades only at discrete, past-dependent times.
• Use a limit order strategy where trades occur when price increments hit predefined thresholds, enabling high-frequency trading without assuming market randomness.
• Embed high-frequency discrete-time Bayesian strategies—previously developed for coin-tossing games—into the continuous-time framework.
• Apply the Bayesian strategy of Kumon, Takemura, and Takeuchi (2008) to discrete-time games, using beta-binomial priors to model beliefs about price movement.
• Express the growth of the investor’s capital using information-theoretic quantities, particularly the Kullback–Leibler divergence between empirical and expected price fluctuations.
• Analyze the asymptotic behavior of capital processes by approximating the Kullback–Leibler divergence in terms of total variation and local fluctuation parameters.

Experimental results

Research questions

• RQ1 Can an investor in a continuous-time market force the variation exponent of the asset price path to be exactly two using only discrete, path-dependent trading?
• RQ2 How does the capital growth of an investor depend on the deviation of the market path from quadratic variation, without assuming stochastic processes?
• RQ3 What is the role of Kullback–Leibler information in quantifying the exponential growth of capital in a game-theoretic framework?
• RQ4 Can high-frequency limit order strategies in discrete time be embedded into continuous-time games to achieve pathwise control over market properties?
• RQ5 Under what conditions does the investor’s capital diverge to infinity, and how does this relate to the Hölder exponent of the price path?

Key findings

• An investor can essentially force the variation exponent of the asset price path to be exactly two by employing a high-frequency limit order strategy based on Bayesian updating.
• Exponential capital growth occurs if the market path has a Hölder exponent H > 0.5 and the local fluctuation |L(T)| ≥ A/4, leading to divergence of the investor's capital to infinity.
• When the total variation of the path is approximately caBk, the capital growth rate is asymptotically governed by terms involving a(1−B)k and L2(T), with the dominant term depending on the ratio of the path's fluctuation to its variation.
• In the case where a1 = a2, the exponential growth part of the capital is described by a(1−B)k / c × L2(T), and this term dominates the growth when H > 0.5 and |L(T)| ≥ A/4.
• When a1 ≠ a2, the capital still diverges to infinity if H > 0.5 and |L(T)| ≥ A/4, and also diverges when H < 0.5, indicating robustness of the strategy across different path regularities.
• The exponential growth of capital is precisely quantified by the Kullback–Leibler divergence between the empirical price fluctuation and the expected behavior under the assumption of quadratic variation, providing a direct link between information theory and financial performance.