  It is well known that mean-variance portfolio selection is a
time-inconsistent optimal control problem in the sense that it does not satisfy
Bellman's optimality principle and therefore the usual dynamic programming
approach fails. We develop a time- consistent formulation of this problem,
which is based on a local notion of optimality called local mean-variance
efficiency, in a general semimartingale setting. We start in discrete time,
where the formulation is straightforward, and then find the natural extension
to continuous time. This complements and generalises the formulation by Basak
and Chabakauri (2010) and the corresponding example in Bj\"ork and Murgoci
(2010), where the treatment and the notion of optimality rely on an underlying
Markovian framework. We justify the continuous-time formulation by showing that
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
optimal strategy in terms of the structure condition and the
F\"ollmer-Schweizer decomposition of the mean-variance tradeoff. As a
byproduct, this also gives new convergence results for the F\"ollmer-Schweizer
decomposition, i.e. for locally risk minimising strategies.
