= Bernstein bound for bounded-jump martingales
{c}
{title2=$\mathbb P(\sup_{s\le t}|M_s|\ge\eta)\le2e^{-\eta^2/[2(v+c\eta/3)]}$}
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 bound
$$
e^{\theta h}-1-\theta h\le\frac{\theta^2h^2}{2(1-\theta c/3)}\qquad(|h|\le c)
$$
makes 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 $k!\ge2\,3^{k-2}$ for $k\ge2$. If $v=0$, the compensated martingale is constant and the crossing probability is zero; the parameter choice above is needed only for $v>0$.
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