Convergence of the embedded mean-variance optimal points with discrete sampling

摘要

A numerical technique based on the embedding technique proposed in (Math Finan 10:387-406, 2000), (Appl Math Optim 42:19-33, 2000) for dynamic mean-variance (MV) optimization problems may yield spurious points, i.e. points which are not on the efficient frontier. In (SIAM J Control Optim 52:1527-1546 2014), 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.