## Journal of Applied Mathematics

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

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

#### Abstract

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

**Source**

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

**Dates**

Received: 30 December 2015

Accepted: 15 March 2016

First available in Project Euclid: 15 June 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.jam/1465996499

**Digital Object Identifier**

doi:10.1155/2016/4543298

**Mathematical Reviews number (MathSciNet)**

MR3500858

#### Citation

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. https://projecteuclid.org/euclid.jam/1465996499