Becoming Exponentially Rich When The Price Process Is Too Volatile

This question belongs to continuous-time game-theoretic probability. It is proved in the new version of [Vovk 2010] (Appendix A, Proposition 7) that the trader can become rich when the $p$-variation of the price process is large for $p>2$. It would be interesting to prove a similar statement for $\log S$ in place of $S$, where $S$ is the trader's capital (and it is likely that the price process $\omega$ will have to be replaced by $\log\omega$). One way of achieving this is to replace Doob's additive argument (used in the proof of Doob's convergence theorem) by its multiplicative counterpart (used in the proof of Levy's zero-one law given in [Shafer et al. 2010]). One danger in using the multiplicative argument is the possibility of losing capital, but we can insure against it by using the methods of [Dawid et al. 2010]. The final inequality for $\log S$ is likely to involve not only ${\rm var}_p(\log\omega)$ but also the final value $\omega(T)$ (we cannot expect a good lower bound on $\log S$ if $\omega(T)$ is very small).

Bibliography

  • A. P. Dawid, Steven de Rooij, Glenn Shafer, Alexander Shen, Nikolai Vereshchagin, and Vladimir Vovk (2010). Insuring against loss of evidence in game-theoretic probability. Working Paper 34.
  • Glenn Shafer, Vladimir Vovk, and Akimichi Takemura (2010). Levy's zero-one law in game-theoretic probability. Working Paper 29.
  • Vladimir Vovk (2010). Rough paths in idealized financial markets. arXiv technical report, May 2010. The new version (15 Dec 2010) is here.