 Martingale decomposition of a density-dependent jump process

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.