Continuous-time trading and the emergence of 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. 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 We also give an elegant statement, inspired by Peter McCullagh's unpublished work, of this result in terms of game-theoretic probability theory.
|Date of creation:||Apr 2009|
|Date of revision:||May 2015|
|Publication status:||Published in Finance and Stochastics 16:561-609 (2012)|
|Contact details of provider:|| Web page: http://arxiv.org/|
References listed on IDEAS
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- Vladimir Vovk, 2007. "Continuous-time trading and emergence of randomness," Papers 0712.1275, arXiv.org, revised Dec 2007.
- Masayuki Kumon & Akimichi Takemura & Kei Takeuchi, 2005. "Capital process and optimality properties of a Bayesian Skeptic in coin-tossing games," Papers math/0510662, arXiv.org, revised Sep 2008.
- V. Vovk, 1993. "Forecasting point and continuous processes: Prequential analysis," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer, vol. 2(1), pages 189-217, December.
- Vladimir Vovk, 2007. "Continuous-time trading and emergence of volatility," Papers 0712.1483, arXiv.org, revised Dec 2007.
- Kei Takeuchi & Masayuki Kumon & Akimichi Takemura, 2007. "A new formulation of asset trading games in continuous time with essential forcing of variation exponent," Papers 0708.0275, arXiv.org, revised Jan 2010.
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