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Journal of Mathematics
Volume 2013, Article ID 726297, 9 pages
Research Article

Mean-Variance Portfolio Selection with Margin Requirements

Yuan Zhou1,2 and Zhe Wu1

1School of Mathematics, Fudan University, Shanghai 200433, China
2Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620-5700, USA

Received 18 November 2012; Revised 23 February 2013; Accepted 24 February 2013

Academic Editor: Cedric Yiu

Copyright © 2013 Yuan Zhou and Zhe Wu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We study the continuous-time mean-variance portfolio selection problem in the situation when investors must pay margin for short selling. The problem is essentially a nonlinear stochastic optimal control problem because the coefficients of positive and negative parts of control variables are different. We can not apply the results of stochastic linearquadratic (LQ) problem. Also the solution of corresponding Hamilton-Jacobi-Bellman (HJB) equation is not smooth. Li et al. (2002) studied the case when short selling is prohibited; therefore they only need to consider the positive part of control variables, whereas we need to handle both the positive part and the negative part of control variables. The main difficulty is that the positive part and the negative part are not independent. The previous results are not directly applicable. By decomposing the problem into several subproblems we figure out the solutions of HJB equation in two disjoint regions and then prove it is the viscosity solution of HJB equation. Finally we formulate solution of optimal portfolio and the efficient frontier. We also present two examples showing how different margin rates affect the optimal solutions and the efficient frontier.