The aim of this Thesis is to to study optimal investment in stocks when the stock is subject to default events and investors have different levels of information. The problem is solved in a general setting by combining a perturbation technique to optimization from Biagini and Øksendal (2005) and default intensities from credit risk literature, in particular Lando (1998).
We assume that the investor has two investment oppurtunities, one riskfree asset and one risky asset subject to default. The dynamics of the risky asset is modelled as a stochastic exponential. The regular random noise in the stochastic exponential is driven by forward integrals with respect to Brownian Motions and Poisson Random measures. The forward integral is an extension of the Itô integral that are suited for modelling insider of partial information for the investor, which is of particular interest to the Thesis. We are also able to state some new results on existence and convergence of forward integrals.
The default is modelled by a jump process with random intensity and only one jump, with dynamics as a doubly stochastic Poisson process.
We then state the necessary and sufficient criteria for a weak form of solutions to the optimal portfolio problem; stochastic controls that are locally maximal. The problem is then solved analytically for the logarithmic utility and we sketch a computable algorithm for other utility functions.