Optimal selection portfolio problem: a semi-linear PDE approach

Abstract

In this paper, we consider the problem of optimal portfolio choice for an investor who wants to maximize the utility of his/her terminal wealth. This work is an overview of the PDE approach for the optimization problem resolution. This approach consists in studying the Hamilton–Jacobi–Bellman equation (HJB equation) associated with the investment problem. In the first part, we consider an investment problem with stochastic volatilities and portfolio constraints. The value function of the investment problem is a viscosity solution of the fully nonlinear HJB equation which can be solved when the risky asset number is low. When the risky asset number is high, the numerical study of the HJB equations is costly. The second section deals with the investment problem with constraints on proportion of the wealth invested in risky assets. This part illustrates the results of Bouzguenda et al. (2009) who studied the backward stochastic differential equations associated with the transformed semi-linear equation and suggested a numerical scheme for the resolution based on the iterative regressions on functions bases and Monte Carlo Method.

Keywords:

• stochastic volatility
• optimal portfolio
• semi-linear partial differential equation
• smooth solution
• backward stochastic differential equation
• regression on function bases

2000 Mathematics Subject Classification::

• 60H30
• 60H10
• 90A09
• 93E20

Notes

^1. [email protected]