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Finance and Stochastics

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01.10.2012

Continuous-time trading and the emergence of probability

verfasst von: Vladimir Vovk

Erschienen in: Finance and Stochastics | Ausgabe 4/2012

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Abstract

This paper establishes a non-stochastic analog 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 paths, 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 We also give an elegant statement, inspired by Peter McCullagh’s unpublished work, of this result in terms of game-theoretic probability theory.

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Titel: Continuous-time trading and the emergence of probability
verfasst von: Vladimir Vovk
Publikationsdatum: 01.10.2012
Verlag: Springer-Verlag
Erschienen in: Finance and Stochastics / Ausgabe 4/2012
Print ISSN: 0949-2984
Elektronische ISSN: 1432-1122
DOI: https://doi.org/10.1007/s00780-012-0180-5

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