  We study optimal investment in an asset subject to risk of default for
investors that rely on different levels of information. The price dynamics can
include noises both from a Wiener process and a Poisson random measure with
infinite activity. The default events are modelled via a counting process in
line with large part of the literature in credit risk. In order to deal with
both cases of inside and partial information we consider the framework of the
anticipating calculus of forward integration. This does not require a priori
assumptions typical of the framework of enlargement of filtrations. We find
necessary and sufficient conditions for the existence of a locally maximizing
portfolio of the expected utility at terminal time. We consider a large class
of utility functions. In addition we show that the existence of the solution
implies the semi-martingale property of the noises driving the stock. Some
discussion on unicity of the maxima is included.
