Convergence of the Embedded Mean-Variance Optimal Points with Discrete Sampling

Abstract

A numerical technique based on the embedding technique proposed in [21, 33] for dynamic mean-variance (MV) optimization problems may yield spurious points, i.e. points which are not on the efficient frontier. In [27], it is shown that spurious points can be eliminated by examining the left upper convex hull of the solution of the embedded problem. However, any numerical algorithm will generate only a discrete sampling of the solution set of the embedded problem. In this paper, we formally establish that, under mild assumptions, every limit point of a suitably defined sequence of upper convex hulls of the sampled solution of the embedded problem is on the original MV efficient frontier. For illustration, we discuss an MV asset-liability problem under jump diffusions, which is solved using a numerical Hamilton-Jacobi-Bellman partial differential equation approach.

Keywords: mean-variance, scalarization optimization, embedding, Pareto optimal, asset-liability, Hamilton-Jacobi-Bellman (HJB) equation, jump diffusion

JEL Classification: E40, E43, G12, G13, C61, C63