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 $(\mathcal V,\mathcal E)$, representing variance $\mathcal V$ and expected value $\mathcal E$, 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 ...