Preservation of Scalarization Optimal Points In

A continuous time mean-variance (MV) problem optimizes the bi-objective criteria 5 (V, E), respectively representing variance V and expected value E of a random variable at the end 6 of a time horizon T . This problem is computationally challenging since the dynamic programming 7 principle cannot be directly applied to the variance criterion. An embedding technique has been 8 proposed in [18, 25] to generate the set of MV scalarization optimal points, which is in general a 9 subset of the mean-variance Pareto optimal points. However, there are a number of complications 10 when we apply the embedding technique in the context of a numerical algorithm. In particular, 11 the frontier generated by the embedding technique may contain spurious points which are not MV 12 optimal. In this paper, we propose a method to eliminate such points, when they exist. We show 13 that the original MV scalarization optimal objective set is preserved if we consider the scalarization 14 optimal points (SOPs) with respect to the MV objective set derived from the embedding technique. 15 Specifically, we establish that these two SOP sets are identical. For illustration, we apply the proposed 16 method to an optimal trade execution problem, which is solved using a numerical Hamilton Jacobi 17 Bellman (HJB) PDE approach. 18