# Continuous-time Game-theoretic Probability

The most developed part of game-theoretic probability deals with discrete time. These are the main approaches to continuous time:

- Shafer and Vovk (2001, Chapter 11) developed a continuous-time theory based on non-standard analysis. This theory is close to the discrete-time theory in that it involves genuine on-line interaction between the main players (Reality, Forecaster, Sceptic). The main disadvantage of this approach is that it is based on an arbitrary choice of an ultrafilter on the set of natural numbers; this makes it aesthetically less appealing.
- Takeuchi et al. (2007) developed an approach based on Takeuchi's technique of "high-frequency limit order strategies" and avoiding non-standard analysis.
- In a series of papers Vovk suggested to widen class of trading strategies used by Takeuchi et al. (2007) to ensure the sigma-subadditivity of upper probability. This led to more compact statements of the main results but re-introduced the requirement of measurability of trading strategies. It remains an open question whether the need for measurability is real: cf. the article "Coherence of game-theoretic Brownian motion".

Other open problems in continuous-time game-theoretic probability:

- Coherence for continuous price processes
- Coherence for discontinuous price processes
- Volatility of discontinuous price processes
- Becoming exponentially rich when the price process is too volatile
- Continuous price processes in the Besov hierarchy

### Bibliography

- Glenn Shafer and Vladimir Vovk.
*Probability and finance: It's only a game!*. New York: Wiley, 2001. - Kei Takeuchi, Masayuki Kumon, Akimichi Takemura. A new formulation of asset trading games in continuous time with essential forcing of variation exponent. arXiv technical report, August 2007.
- Vladimir Vovk. Continuous-time trading and the emergence of randomness. arXiv technical report, December 2007.
- Vladimir Vovk. Continuous-time trading and the emergence of volatility. arXiv technical report, December 2007.
- Vladimir Vovk. Game-theoretic Brownian motion. arXiv technical report, January 2008.
- Vladimir Vovk. Continuous-time trading and the emergence of probability. arXiv technical report, August 2010.
- Vladimir Vovk. Rough paths in idealized financial markets. arXiv technical report, May 2010.