 
Summary: INITIAL ENLARGEMENT OF FILTRATIONS AND
ENTROPY OF POISSON COMPENSATORS
STEFAN ANKIRCHNER AND JAKUB ZWIERZ
Abstract. Let µ be a Poisson random measure, F the smallest filtra
tion satisfying the usual conditions and containing the one generated by
µ, and let G be the initial enlargement of F with the field generated
by a random variable G. In this paper, we first show that the mutual
information between the enlarging random variable G and the algebra
generated by the Poisson random measure µ is equal to the expected rel
ative entropy of the Gcompensator relative to the Fcompensator of the
random measure µ. We then use this link to gain some insight into the
changes of DoobMeyer decompositions of stochastic processes when the
filtration is enlarged from F to G. In particular, we show that if the mu
tual information between G and the algebra generated by the Poisson
random measure µ is finite, then every square integrable Fmartingale is
a Gsemimartingale that belongs to the normed space S1
relative to G.
Introduction
The perception of a random process changes if events of its future de
velopment are anticipated. Such anticipations can be modeled in terms
