Before and after default: information and optimal portfolio via anticipating calculus
Abstract
Default risk calculus emerges naturally in a portfolio optimization problem when the risky asset is threatened with a bankruptcy. The usual stochastic control techniques do not hold in this case and some additional assumptions are generally added to achieve the optimization in a beforeandafter default context. We show how it is possible to avoid one of theses restrictive assumptions, the socalled Jacod density hypothesis, by using the framework of the forward integration. In particular, in the logarithmic utility case, in order to get the optimal portfolio the right condition it is proved to be the intensity hypothesis. We use the anticipating calculus to analyze the existence of the optimal portfolio for the logarithmic utility, and than under the assumption of existence of the optimal portfolio we prove the semimartingale decomposition of the risky asset in the filtration enlarged with the default process.
 Publication:

arXiv eprints
 Pub Date:
 July 2022
 arXiv:
 arXiv:2208.07163
 Bibcode:
 2022arXiv220807163S
 Keywords:

 Quantitative Finance  Portfolio Management;
 Mathematics  Optimization and Control;
 Mathematics  Probability;
 60H0;
 60G44;
 93E20