On the representation of integral-valued random measures and local martingales by means of random measures with deterministic compensators

The relation $\mu(\omega;A)=p(\omega;\psi^{-1}_\omega(A))$ between integral-valued measures $\mu(\omega;\cdot\,)$ and $p(\omega;\cdot\,)$ and the compensators $\nu(\omega;\cdot\,)$ and $q(\,\cdot\,)$, respectively, is established ($q$ is a deterministic measure), where $\psi_\omega(\,\cdot\,)$ is a predictable mapping, provided that $\nu(\omega;A)=q(\psi^{-1}_\omega(A))$. This result is used to represent a local martingale in the form of a sum of stochastic integrals with respect to a continuous Gaussian martingale and the martingale measure $p-q$.
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