Time-Consistent Mean-Variance Portfolio Selection inDiscrete and Continuous Time

It is well known that mean-variance portfolio selection is a time-inconsistent optimalcontrol problem in the sense that it does not satisfy Bellman’s optimalityprinciple and therefore the usual dynamic programming approach fails. We developa time-consistent formulation of this problem, which is based on a local notion ofoptimality called local mean-variance efficiency, in a general semimartingale setting.We start in discrete time, where the formulation is straightforward, and then findthe natural extension to continuous time. This generalises recent results by Basakand Chabakauri (2010), Bj¨ork and Murgoci (2008) and Bj¨ork, Murgoci and Zhou(2010), where the treatment as well as the notion of optimality rely on an underlyingMarkovian framework. We justify the continuous-time formulation by showingthat it coincides with the continuous-time limit of the discrete-time formulation.The proof of this convergence is based on a global description of the locally optimalstrategy in terms of the structure condition and the F¨ollmer–Schweizer decompositionof the mean-variance tradeoff process. As a byproduct, this also gives newconvergence results for the F¨ollmer–Schweizer decomposition, i.e. for locally riskminimising strategies.[...]