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

Abstract

The embedding technique proposed in [13, 22] for mean-variance (MV) optimization problems may yield spurious points. These are points in the MV objective set, derived from the embedding technique, but are not MV scalarization optimal points (SOPs) with respect to this set. In [17], it is shown that a spurious point is the point at which a supporting hyperplane for the embedded MV objective set does not exist. In addition, it is shown that the resulting set, obtained after eliminating spurious points from the embedded MV objective set, is identical to the set of original MV scalarization optimal objectives [17]. In numerical computation, however, significant complexities remain. This is due to the fact that it is only possible to obtain a subset of the computed MV embedded objective set, with each element corresponding to a solution for a single sampled embedding parameter value. As a result, an important question is whether or not, for a sufficiently large number of sampled embedding parameters, the set of SOPs, with respect to the afore-mentioned finite subset of the computed MV embedded objective set, can sufficiently well approximate the SOPs with respect to the entire computed MV set with the embedding parameter in (-∞, ∞). In this paper, we formally establish that, under mild assumptions, every limit point of a SOP sequence, indexed by the embedding parameter sampling level, is a SOP of the computed MV embedded objective set for all embedding parameters. 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.