Volatility Of Discontinuous Price Processes

This question belongs to continuous-time game-theoretic probability. The sample space $\Omega$ is the set of all positive right-continuous functions of time.

  • Let $\psi$ be Taylor's function: $\psi(u):=u^2\slash{}(2\ln^*\ln^*u)$, where $\ln^*u:=\max(1,|\ln u|)$. Is it true that $v_{\psi}<\infty$ almost surely? It is known (see [Vovk 2010]) that the answer is "yes" when $\Omega$ is the set of all continuous functions of time. In the case of discontinuous functions, it is only known that $v_{\phi}<\infty$, where $\phi(u):=(u\slash{}\ln^*u)^2$ [Vovk 2010].