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 2009 *Bernoulli*
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:

- Continuous-time trading
and the emergence of randomness
(last revised June 2008).
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 paths (assumed continuous). Published in

*Stochastics*. - Continuous-time trading
and emergence of volatility
(December 2007).
This paper shows that the strong variation exponent of non-constant continuous price paths has to be 2, as in the case of Brownian motion. Published in

*Electronic Communications in Probability*. - Game-theoretic Brownian motion
(January 2008).
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.

- Continuous-time trading
and the emergence of probability
(last revised July 2011).
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. To appear in

*Finance and Stochastics*. - Rough paths
in idealized financial markets
(last revised May 2011).
This paper partially extends the result of paper 2 by showing that the variation of right-continuous positive price paths cannot exceed 2 (i.e., be much rougher than Brownian motion). A shorter version is published in

*Lithuanian Mathematical Journal*. - Ito calculus without probability
in idealized financial markets
(August 2011).
This paper proves the existence of quadratic variation for cadlag price paths with a mild limitation on the size of jumps.

- The efficient index hypothesis
and its implications in the BSM model
(September 2011).
An index I is supposed to be a tradable security and to satisfy the Black-Scholes-Merton model dI(t)/I(t)=μdt+σdW(t). It is shown that the appreciation rate μ has to be close to r+σ², where r is the interest rate (assumed constant): if it is not, the index can be outperformed greatly. This gives an equity premium of σ².

- The Capital Asset Pricing Model
as a corollary of the Black-Scholes model
(September 2011).
Considering a market containing a stock and an index, this paper shows that, for a long investment horizon, the appreciation rate μ of the stock has to be close to the interest rate plus the covariance between the volatility vectors of the stock and the index. (If it is not, the index can be outperformed greatly.) This contains both a version of the Capital Asset Pricing Model and the result of paper 7 that the equity premium is close to the squared volatility of the index.