Predictable compensator of a jump measure ($\mathbb{E}(H\ast N)=\mathbb{E}(H\ast \nu )(H\ge 0\text{ predictable})$)

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.