Jump measure of a Markov jump process ($\mu ={\sum }_{t:X_t\ne X_{t-}}{\delta }_{(t,X_t)},\nu (dt,dy)=q(X_{t-},dy)dt$)

For a Markov jump process with off-diagonal rate kernel $q(x,dy)$, the jump measure records the jump time and the destination state. Relative to a filtration for which the process has the Markov jump property, its predictable compensator is the displayed rate measure, up to explosion. The corresponding generator is $Qf(x)=\int [f(y)-f(x)]q(x,dy)$. An enlarged filtration revealing future jumps need not preserve this compensator, so the filtration condition is essential.

The predictable random measure $\nu$ matching the expected integrals of the jump measure $N$ against every nonnegative predictable integrand. For displacement marks of a Markov jump process, $\nu (ds,dy)=K(X_{s-},dy)ds$. Multiplying the integrand by an event known before an interval proves that the compensated integral has conditionally centered increments; expected absolute integrability makes it a true martingale. Squared-integrand integrability gives the square-integrable compensation isometry.

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.