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".