Journal of Applied Mathematics

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

Moussa Kounta

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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 V 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 V 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.

Article information

J. Appl. Math., Volume 2016 (2016), Article ID 4543298, 14 pages.

Received: 30 December 2015
Accepted: 15 March 2016
First available in Project Euclid: 15 June 2016

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Kounta, Moussa. Viscosity Solution of Mean-Variance Portfolio Selection of a Jump Markov Process with No-Shorting Constraints. J. Appl. Math. 2016 (2016), Article ID 4543298, 14 pages. doi:10.1155/2016/4543298.

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  • H. M. Markowitz, “Portfolio selection,” The Journal of Finance, vol. 7, no. 1, pp. 77–91, 1952.
  • D. Li and W.-L. Ng, “Optimal dynamic portfolio selection: multi-period mean variance formulation,” Mathematical Finance, vol. 10, no. 3, pp. 387–406, 2000.
  • X. Y. Zhou and D. Li, “Continuous-time mean-variance portfolio selection: a stochastic LQ framework,” Applied Mathematics and Optimization, vol. 42, no. 1, pp. 19–33, 2000.
  • X. Y. Zhou and G. Yin, “Markowitz's mean-variance portfolio selection with regime switching: a continuous-time model,” SIAM Journal on Control and Optimization, vol. 42, no. 4, pp. 1466–1482, 2003.
  • X. Li, X. Y. Zhou, and A. E. Lim, “Dynamic mean-variance portfolio selection with no-shorting constraints,” SIAM Journal on Control and Optimization, vol. 40, no. 5, pp. 1540–1555, 2002.
  • G.-L. Xu and S. E. Shreve, “A duality method for optimal consumption and investment under short-selling prohibition. I. General market coefficients,” The Annals of Applied Probability, vol. 2, no. 1, pp. 87–112, 1992.
  • M. G. Crandall and P.-L. Lions, “Viscosity solutions of Hamilton-Jacobi equations,” Transactions of the American Mathematical Society, vol. 277, no. 1, pp. 1–42, 1983.
  • W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity, Springer, 2nd edition, 2006.
  • S. M. Lenhart, “Viscosity solutions for weakly coupled systems of first-order partial differential equations,” Journal of Mathematical Analysis and Applications, vol. 131, no. 1, pp. 180–193, 1988.
  • S. Blanes, F. Casas, J. A. Oteo, and J. Ros, “The Magnus expansion and some of its applications,” Physics Reports, vol. 470, no. 5-6, pp. 151–238, 2009.
  • S. Blanes and E. Ponsoda, “Time-averaging and exponential integrators for non-homogeneous linear IVPs and BVPs,” Applied Numerical Mathematics, vol. 62, no. 8, pp. 875–894, 2012.
  • R. C. Merton, “An analytical derivation of the efficient portfolio frontier,” The Journal of Financial and Quantitative Analysis, vol. 7, no. 4, pp. 1851–1872, 1972. \endinput