Lévy kernel of a Markov jump process ($K(x,dz)$)

A kernel specifying the jump rate from state $x$ for each displacement $z$. With the displacement convention the infinitesimal generator is $L\phi (x)=\int (\phi (x+z)-\phi (x))K(x,dz)$. The destination convention instead records a rate measure on the new state, translating each displacement by $x$. For finite total rate, that total is the exponential holding rate and the normalized measure is the next-jump distribution.

With the displacement convention for a Lévy kernel of a Markov jump process, suppose its total rate is bounded, its first moment is zero and its second moment $v$ is bounded by $V$. The bounded rate prevents explosion. The compensated displacement integral is a square-integrable martingale with $\mathbb{E}{X}_t^2\le Vt$. The displayed process is the sum of $2\int X_{s-}d{X}_s$ and the compensated sum of squared jumps. The first term is square integrable since $4\mathbb{E}{\int }_0^t{X}_{s-}^{2}v(X_{s-})ds\le 2V^2{t}^2$; the second is integrable because its expected total jump-square sum is at most $Vt$. Consequently the square compensation is a true martingale, without a fourth-moment assumption.