标题： 离散与连续时间下时间一致的均值-方差投资组合选择 显示英文标题

标题： Time-Consistent Mean-Variance Portfolio Selection in Discrete and Continuous Time

摘要： 众所周知，均值-方差投资组合选择是一个本质上具有时间不一致性的问题，因为它不满足Bellman最优性原理，因此通常的动态规划方法失效。我们在一般半鞅设定下，发展了这一问题的一个时间一致的表述，该表述基于一种局部意义上的最优性概念——局部均值-方差有效性。我们从离散时间开始，其中表述是直接明了的，然后找到其到连续时间的自然扩展。这补充和推广了Basak和Chabakauri（2010）的工作及其在Björk和Murgoci（2010）中的相应例子，在这些工作中处理方式及最优性的概念依赖于潜在的马尔可夫框架。我们通过证明连续时间表述与离散时间表述的连续时间极限相吻合来验证连续时间表述的合理性。这一收敛性的证明基于局部最优策略的结构条件以及均值-方差权衡的Föllmer-Schweizer分解的全局描述。作为副产品，这也给出了新的Föllmer-Schweizer分解的收敛结果，即关于局部风险最小化策略的结果。 显示英文摘要

摘要： 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.