The background for the general mathematical link between utility and information
theory investigated in this paper is a simple financial market model with two kinds of small
traders: less informed traders and insiders whose extra information is represented by an
enlargement of the other agents' filtration. The expected logarithmic utility increment,
i.e. the difference of the insider's and the less informed trader's expected logarithmic
utility is described in terms of the information drift, i.e. the drift one has to eliminate
in order to perceive the price dynamics as a martingale from the insider's perspective.
On the one hand, we describe the information drift in a very general setting by natural
quantities expressing the probabilistic better informed view of the world. This on the
other hand allows us to identify the additional utility by entropy related quantities known
from information theory. In particular in a complete market in which the insider has some
fixed additional information during the entire trading interval, its utility increment can
be represented by the Shannon information of his extra knowledge. For general markets,
and in some particular examples, we provide estimates of maximal utility by information
inequalities.
Let (Gt) be an enlargement of the filtration (Ft). Jeulin and Jacod
discussed a sufficient criterion for the inheritance of the semimartingale property
when passing to the larger filtration. We provide alternative proofs of their results
in a more general setting by using decoupling measures and Girsanov's changes of
measure. We derive necessary and sufficient conditions for the embedding of vector
spaces of (Ft)-semimartingales into spaces of (Gt)-semimartingales to be continuous
in terms of generalized entropies of the information increment.