Preservation of Scalarization Optimal Points in the Embedding Technique for Continuous Time Mean Variance Optimization

A continuous time mean variance (MV) problem optimizes the biobjective criteria (V , E), representing variance V and expected value E, respectively, of a random variable at the end of a time horizon T . This problem is computationally challenging since the dynamic programming principle cannot be directly applied to the variance criterion. An embedding technique has been proposed in [D. Li and W. L. Ng, Math. Finance, 10 (2000), pp. 387–406; X. Y. Zhou and D. Li, Appl. Math. Optim., 42 (2000), pp. 19–33] to generate the set of MV scalarization optimal points, which is in general a subset of the MV Pareto optimal points. However, there are a number of complications when we apply the embedding technique in the context of a numerical algorithm. In particular, the frontier generated by the embedding technique may contain spurious points which are not MV optimal. In this paper, we propose a method to eliminate such points, when they exist. We show that the original MV scalarization optimal objective set is preserved if we consider the scalarization optimal points (SOPs) with respect to the MV objective set derived from the embedding technique. Specifically, we establish that these two SOP sets are identical. For illustration, we apply the proposed method to an optimal trade execution problem, which is solved using a numerical Hamilton–Jacobi–Bellman PDE approach.