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

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