# Continuous-time trading and the emergence of probability

*Vladimir Vovk* ()

*Finance and Stochastics*, 2012, vol. 16, issue 4, 561-609

**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 We also give an elegant statement, inspired by Peter McCullagh’s unpublished work, of this result in terms of game-theoretic probability theory. Copyright Springer-Verlag 2012

**Keywords:** Game-theoretic probability; Continuous time; Emergence of probability; Continuous price paths; Incomplete markets; 91G99; 60G17; 60G05; 60G44; C58; G13; G14 (search for similar items in EconPapers)

**Date:** 2012

**References:** View references in EconPapers View complete reference list from CitEc

**Citations:** View citations in EconPapers (12) Track citations by RSS feed

**Downloads:** (external link)

http://hdl.handle.net/10.1007/s00780-012-0180-5 (text/html)

Access to full text is restricted to subscribers.

**Related works:**

This item may be available elsewhere in EconPapers: Search for items with the same title.

**Export reference:** BibTeX
RIS (EndNote, ProCite, RefMan)
HTML/Text

**Persistent link:** https://EconPapers.repec.org/RePEc:spr:finsto:v:16:y:2012:i:4:p:561-609

**Ordering information:** This journal article can be ordered from

http://www.springer. ... ance/journal/780/PS2

Access Statistics for this article

Finance and Stochastics is currently edited by *M. Schweizer*

More articles in Finance and Stochastics from Springer

Bibliographic data for series maintained by Sonal Shukla ().