= Jump measure of a Markov jump process
{title2=$\mu=\sum_{t:\,X_t\ne X_{t-}}\delta_{(t,X_t)},\quad\nu(dt,dy)=q(X_{t-},dy)dt$}
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.
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