= Martingale decomposition of a density-dependent jump process
{title2=$Z_t^N=Z_0^N+\int_0^tF(Z_s^N)ds+M_t^N$}
With jumps $\ell/N$ at rates $N\beta_\ell(z)$, compensate each jump count by its integrated rate. The drift is $F(z)=\sum_\ell\ell\beta_\ell(z)$ and the <predictable quadratic variation> matrix of the <martingale> term is $N^{-1}\int_0^t\sum_\ell\ell\ell^{\mathsf T}\beta_\ell(Z_s^N)ds$. On a compact region with bounded rates this bracket is of order $1/N$. A Lipschitz drift and the <Gronwall inequality> transfer martingale concentration to the <fluid limit>.
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