On Lower and Upper Functions for Square Integrable Martingales

We consider a locally square integrable martingale $M = (M_t,\mathcal F_t)_{t\ge 0}$ satisfying $\lim _{t\to\infty }\langle M\rangle _t = +\infty$ ($\mathsf P$-a.s.), with predictably bounded jumps $|\Delta M_s| \le g(\langle M\rangle_s)$ for $s\ge t_0\ge 0$, where $g$ is a nonnegative nondecreasing continuous function and $\langle M \rangle$ is the predictable quadratic characteristic of $M$. For a nonnegative nondecreasing continuous function $\phi$, we give a sufficient condition similar to the Kolmogorov–Petrovskii test saying when $\phi (\langle M\rangle)$ is a lower function for $|M|$. In particular, if $\phi(t)=\sqrt{2t\ln\ln t}$ and $g(t)=O({t^{1/2}}/{ (\ln t)^{1+\delta}})$, we obtain that $\sqrt {2\langle M\rangle \ln\ln \langle M\rangle _t}$ is lower for $|M|$ and $\limsup _{t\to \infty} {|M_t|}/ {\sqrt { 2\langle M\rangle \ln\ln \langle M\rangle _t}}\ge 1$ $\mathsf P$-a.s. If the predictable quadratic characteristic $\langle M\rangle$ is continuous in $t$, then, under some supplementary conditions on jumps of $M$, we prove an analogous result for $\phi (t) = \sqrt {2t\ln\ln t}$ and $g(t)=O (t^{1/2}/(\ln \ln t)^{3/2})$.