Optimal portfolio and consumption selection with default risk

Abstract

We investigate an optimal portfolio and consumption choice problem with a defaultable security. Under the goal of maximizing the expected discounted utility of the average past consumption, a dynamic programming principle is applied to derive a pair of second-order parabolic Hamilton-Jacobi- Bellman (HJB) equations with gradient constraints. We explore these HJB equations by a viscosity solution approach and characterize the post-default and pre-default value functions as a unique pair of constrained viscosity solutions to the HJB equations.

Keywords

Defaultable security / average past consumption / Hamilton-Jacobi- Bellman (HJB) equation / post(pre)-default / constrained viscosity solution

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