Before and after default: information and optimal portfolio via anticipating calculus

TL;DR

This paper introduces a novel approach using forward integration to optimize portfolios under default risk, overcoming limitations of traditional methods and providing insights for risk management strategies.

Contribution

It proposes an alternative to stochastic control by employing forward integration, relaxing assumptions like the Jacod density hypothesis, and characterizing the asset dynamics under default risk.

Findings

We identify the weaker intensity hypothesis as sufficient for optimality with logarithmic utility.

The semimartingale decomposition of assets is established in an enlarged filtration.

The approach enhances risk management strategies in default scenarios.

Abstract

Default risk calculus plays a crucial role in portfolio optimization when the risky asset is under threat of bankruptcy. However, traditional stochastic control techniques are not applicable in this scenario, and additional assumptions are required to obtain the optimal solution in a before-and-after default context. We propose an alternative approach using forward integration, which allows to avoid one of the restrictive assumptions, the Jacod density hypothesis. We demonstrate that, in the case of logarithmic utility, the weaker intensity hypothesis is the appropriate condition for optimality. Furthermore, we establish the semimartingale decomposition of the risky asset in the filtration that is progressively enlarged to accommodate the default process, under the assumption of the existence of the optimal portfolio. This work aims to provide valueable insights for developing effective risk management strategies when facing default risk.