Initial enlargement of filtrations and entropy of Poisson compensators (Q633140)

Consider a Poisson random measure \(\mu\), denote by \(\mathbb{F}\) the smallest initial filtration satisfying the usual conditions and containing the one generated by \(\mu\), and let \(\mathbb{G}\) be the initial enlargement of \(\mathbb{F}\) with the \(\sigma\)-field generated by a random variable \(G\). The authors show that the mutual information between the enlarging random variable \(G\) and the \(\sigma\)-algebra generated by the Poisson random measure \(\mu\) equals the expected relative entropy of the \(\mathbb{G}\)-compensator relative to the \(\mathbb{F}\)-compensator of the random measure \(\mu\). This result provides some insight into how the Doob-Meyer decomposition of a stochastic process changes when the filtration is enlarged. In particular, the authors prove that, if \(\mathbb{G}\) is an initial enlargement of \(\mathbb{F}\) by some random variable \(G\) with finite entropy, then every square integrable \((\mathbb{F},\mathbb{P})\)-martingale remains a \((\mathbb{G},\mathbb{P})\)-semimartingale and has an information drift that is integrable.