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

TL;DR

This paper presents a novel continuous-time asset trading game framework where an investor can force the asset price path to have a variation exponent of exactly two, linking capital growth to information measures.

Contribution

It introduces a new formulation of asset trading games in continuous time, demonstrating how to control the variation exponent and relate capital growth to information quantities.

Findings

Investor can force the variation exponent to be exactly two.

Capital growth is characterized by Kullback-Leibler information.

Embedding high-frequency discrete games into continuous time is effective.

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.