Utility maximization under trading constraints with discontinuous utility

This paper investigates a utility maximization problem in a Black–Scholes market, in which trading is subject to a convex cone constraint and the utility function is not necessarily continuous or concave. The problem is initially formulated as a stochastic control problem, and a partial differential equation method is subsequently used to study the associated Hamilton–Jacobi–Bellman equation. The value function is shown to be discontinuous at maturity (with the exception of trivial cases), and its lower-continuous envelope is shown to be concave before maturity. The comparison principle shows that the value function is continuous and coincides with that of its concavified problem.