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

TL;DR

This paper develops an HJB framework for solving a broad class of continuous-time mean-variance stochastic control problems, linking viscosity and Sobolev solutions to derive optimal strategies.

Contribution

It introduces a novel approach transforming the mean-variance problem into static and dynamic parts, analyzing the value function via degenerate HJB equations in viscosity and Sobolev senses.

Findings

Establishes existence of solutions to the HJB equation under certain conditions

Connects viscosity and Sobolev solutions for the value function

Provides implications for the optimality of control strategies

Abstract

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 viscosity or in 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.