= Centered finite-rate jump kernel gives square martingales
{title2=$X_t^2-\int_0^tv(X_s)ds$}
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 $\mathbb EX_t^2\le Vt$. The displayed process is the sum of $2\int X_{s-}dX_s$ and the compensated sum of squared jumps. The first term is square <integrable> since $4\mathbb E\int_0^tX_{s-}^2v(X_{s-})ds\le2V^2t^2$; 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.
Back to article page
