the compensator of the jump measure

The compensator is the deterministic (more generally, predictable) drift you must subtract from the random jump measure to turn it into a martingale. Where the Poisson random measure N counts jumps, its compensator gives the average rate of jumping; the difference, the compensated measure, has mean zero and is the martingale that makes the stochastic calculus of jumps work.

For a Lévy process the jump measure N(dt, dx) has compensator (also called the predictable / dual-predictable projection) equal to dt nu(dx): the expected number of jumps in (s, t] x B is exactly (t-s) nu(B). The compensated jump measure is the signed measure N-tilde(dt, dx) = N(dt, dx) - dt nu(dx). For any test function f integrable against the compensator, the process t -> integral over (0,t] x B of f(x) N-tilde(ds, dx) is a martingale with mean zero; this is the precise sense in which the compensator 'centers' the jumps. The Lévy-Itô decomposition's small-jump term is exactly integral of x N-tilde over |x| < 1, which exists as an L^2 martingale because integral over |x|<1 of x^2 nu(dx) < infinity even when the raw integral integral x N diverges. For general semimartingales the compensator is a genuinely random predictable measure (the Doob-Meyer / Grigelionis characteristics), but for Lévy processes it is the deterministic dt nu(dx) thanks to stationary independent increments.

The compensator is what makes martingale methods available for jump processes: it provides the predictable quadratic variation, it is the object in the Itô formula's jump correction, and subtracting it is the only way to integrate the infinitely many small jumps. The crucial honesty: the raw jump measure N is NOT a martingale (it only increases); only the compensated N-tilde is. And the compensator must be PREDICTABLE — for Lévy processes it happens to be deterministic, but in general predictability (not mere adaptedness) is what makes the compensation and the martingale property well posed.