  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. We consider an idealized financial security with continuous
price path, without making any stochastic assumptions. It is shown that typical
price paths possess quadratic variation, where "typical" is understood in the
following game-theoretic sense: there exists a trading strategy that earns
infinite capital without risking more than one monetary unit if the process of
quadratic variation does not exist. Replacing time by the quadratic variation
process, we show that the price path becomes Brownian motion. This is
essentially the same conclusion as in the Dubins-Schwarz result, except that
the probabilities (constituting the Wiener measure) emerge instead of being
postulated. We also give an elegant statement, inspired by Peter McCullagh's
unpublished work, of this result in terms of game-theoretic probability theory.
