An HJB Approach to a General Continuous-Time Mean-Variance Stochastic Control Problem

Abstract

A general continuous mean-variance problem is considered where the cost functional has an integral and a terminal-time component. The problem is transformed into a superposition of a static and a dynamic optimization problems. The value function of the latter can be considered as the solution to a degenerate HJB equation either in viscosity or in Sobolev sense (after regularization) under suitable assumptions and with implications with regards to the optimality of strategies.