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Journal of Applied Mathematics
Volume 2016 (2016), Article ID 4543298, 14 pages
Research Article

Viscosity Solution of Mean-Variance Portfolio Selection of a Jump Markov Process with No-Shorting Constraints

The College of the Bahamas, School of Mathematics, Physics and Technology, P.O. Box 4912, Nassau, Bahamas

Received 30 December 2015; Accepted 15 March 2016

Academic Editor: Jinde Cao

Copyright © 2016 Moussa Kounta. 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 consider the so-called mean-variance portfolio selection problem in continuous time under the constraint that the short-selling of stocks is prohibited where all the market coefficients are random processes. In this situation the Hamilton-Jacobi-Bellman (HJB) equation of the value function of the auxiliary problem becomes a coupled system of backward stochastic partial differential equation. In fact, the value function often does not have the smoothness properties needed to interpret it as a solution to the dynamic programming partial differential equation in the usual (classical) sense; however, in such cases can be interpreted as a viscosity solution. Here we show the unicity of the viscosity solution and we see that the optimal and the value functions are piecewise linear functions based on some Riccati differential equations. In particular we solve the open problem posed by Li and Zhou and Zhou and Yin.