 Bernstein bound for bounded-jump martingales

For a compensated jump martingale starting at zero, with jumps bounded in absolute value by $c$ and predictable quadratic variation at most $v$ on the interval, the displayed maximal bound holds. For $0<\theta c<3$, the elementary exponential boundmakes the exponential of $\theta M$ minus that multiple of its predictable bracket a nonnegative supermartingale. Stop at the first crossing of $\eta$ and use its mean bound. Taking $\theta =\eta /(v+c\eta /3)$ yields the one-sided bound; apply the same argument to $-M$ and use the union bound for the displayed two-sided version. The elementary estimate follows from the exponential series and $k!\ge 23^{k-2}$ for $k\ge 2$. If $v=0$, the compensated martingale is constant and the crossing probability is zero; the parameter choice above is needed only for $v>0$.