Compensated jump-measure local martingale ($H\ast (\mu -\nu )$)

A predictable mark-dependent integrand $H(t,y)$ gives a true martingale ${\int }_0^t\int H(s,y)(\mu -\nu )(ds,dy)$ whenever $\mathbb{E}{\int }_0^t\int |H|d\nu <\infty$ on every finite horizon. Local versions follow when this expected integrability holds after localization. The compensator identity applied to predictable indicators from the past proves that every increment has conditional expectation zero. Choosing $H(s,y)=f(y)-f(X_{s-})$ gives the generator martingale $f(X_t)-f(X_0)-{\int }_0^{t}Qf(X_s)ds$ under the same local integrability conditions.