Constrained Dynamic Mean-Variance Portfolio Selection in Continuous-Time

Weiping Wu,Shan Pang,Ruobing Xue,Lifen Wu

doi:10.3390/a14080252

Abstract

This paper revisits the dynamic MV portfolio selection problem with cone constraints in continuous-time. We first reformulate our constrained MV portfolio selection model into a special constrained LQ optimal control model and develop the optimal portfolio policy of our model. In addition, we provide an alternative method to resolve this dynamic MV portfolio selection problem with cone constraints. More specifically, instead of solving the correspondent HJB equation directly, we develop the optimal solution for this problem by using the special properties of value function induced from its model structure, such as the monotonicity and convexity of value function. Finally, we provide an example to illustrate how to use our solution in real application. The illustrative example demonstrates that our dynamic MV portfolio policy dominates the static MV portfolio policy.

Highlights

The classical static mean-variance (MV) model was pioneered by Markowitz [1] more than sixty years ago, which laid the foundation of modern financial theory
Instead of solving the correspondent HJB equation directly, we developed the optimal solution for this problem by using the special properties of value function induced from its model structure, such as the monotonicity and convexity of value function
This alternative method offers a new way of thinking under the constrained mean-risk framework and can be extended to solve another portfolio selection models with cone constraints

Summary

Introduction

The classical static mean-variance (MV) model was pioneered by Markowitz [1] more than sixty years ago, which laid the foundation of modern financial theory. As for the continuous-time versions of problems of this type, Hu and Zhou [17] solved them by using the backward stochastic differential equation approach They introduced two extended stochastic Riccati equations (ESREs) and characterized the optimal portfolio policy by using the solutions of these two ESREs. their work did not demonstrate how to construct these two ESREs by using the special structure of a model rather than by suspecting them. Instead of solving the correspondent HJB equation directly, we developed the optimal solution for this problem by using the special properties of value function induced from its model structure, such as the monotonicity and convexity of value function This alternative method offers a new way of thinking under the constrained mean-risk framework and can be extended to solve another portfolio selection models with cone constraints. All lemmas and theorems have been proofed in the Appendix A

Problem Formulation

An Illustrative Example

Conclusions