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INITIAL ENLARGEMENT OF FILTRATIONS AND ENTROPY OF POISSON COMPENSATORS
 

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 G-compensator relative to the F-compensator of the
random measure µ. We then use this link to gain some insight into the
changes of Doob-Meyer 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 F-martingale is
a G-semimartingale 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

  

Source: Ankirchner, Stefan - Institut für Angewandte Mathematik, Universität Bonn

 

Collections: Mathematics