Robust portfolio optimization via solution to the Hamilton–Jacobi–Bellman equation

Abstract

We consider a problem of dynamic stochastic portfolio optimization modelled by a fully non-linear Hamilton–Jacobi–Bellman (HJB) equation. Using the Riccati transformation, the HJB equation is transformed to a simpler quasi-linear partial differential equation. An auxiliary quadratic programming problem is obtained, which involves a vector of expected asset returns and a covariance matrix of the returns as input parameters. Since this problem can be sensitive to the input data, we modify the problem from fixed input parameters to worst-case optimization over convex or discrete uncertainty sets both for asset mean returns and their covariance matrix. Qualitative as well as quantitative properties of the value function are analysed along with providing illustrative numerical examples. We show application to robust portfolio optimization for the German DAX30 Index.

Keywords:

• Hamilton–Jacobi–Bellman equation
• robust portfolio optimization
• optimal investment problem

2010 AMS Subject Classifications:

• 90C22
• 91B70
• 70H20
• 35K55
• 65M08

Acknowledgements

This research was supported by the project 7FP EU STRIKE-Computational finance, 304617 and VEGA 1/2429/12.