The conditions are as given in the title. I want to show, for a discrete martingale $M$ with $E[M_0]=0$ and $E[M^2]<\infty$, $P\left(\max_{0\leq s\leq t} M_s > x\right) \leq \frac{E (M_t^2)}{E (M_t^2) + x^2}$.
Kolmogorov gives us $P\left(\max_{0\leq s\leq t} M_s > x\right) \leq \frac{E (M_t^2)}{x^2}.$ I would assume stopping times are involved, but I really can't think of how that extra $E[M_t^2]$ can be added in the denominator.