# 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 -variation of the price process is large for . It would be interesting to prove a similar statement for in place of , where is the trader's capital (and it is likely that the price process will have to be replaced by ). 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 is likely to involve not only but also the final value (we cannot expect a good lower bound on if 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.