 Jump measure of a Markov jump process

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.