This project is part of the Game-Theoretic Probability and Finance project. Treatment of continuous time in Glenn Shafer's and my 2001 book and several subsequent papers was based on non-standard analysis. In their recent (August 2007) paper, Kei Takeuchi, Masayuki Kumon, and Akimichi Takemura suggested an approach avoiding non-standard analysis and introduced in game-theoretic probability the important technique of "high-frequency limit order strategies". (My old 1993 paper published in Test also suggested a game-theoretic approach to continuous-time processes without non-standard analysis, but it was awkward in some respects and so was not pursued further.) The following papers develop the Takeuchi-Kumon-Takemura approach:
A new definition of events of game-theoretic probability zero in continuous time is proposed and used to prove results suggesting that trading in financial markets results in the emergence of properties usually associated with randomness. This paper concentrates on "qualitative" results, stated in terms of order (or order topology) rather than in terms of the precise values taken by the price processes (assumed continuous).
This paper shows that the strong variation exponent of non-constant continuous price processes has to be 2, as in the case of Brownian motion.
This paper suggests a perfect-information game, along the lines of Lévy's characterization of Brownian motion, that formalizes the process of Brownian motion in game-theoretic probability.
This paper establishes a non-stochastic analogue of the celebrated result by Dubins and Schwarz about reduction of continuous martingales to Brownian motion via time change. It contains the main results of papers 1 and 2 as special cases.