Time-consistent mean-variance portfolio selection in discrete and continuous time
Time-consistent mean-variance portfolio selection in discrete and continuous time
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DOI：
10.1007/s00780-012-0189-9
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发表时间：
2012-05
影响因子：
1.7
通讯作者：
Christoph Czichowsky
中科院分区：
文献类型：
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作者：
Christoph Czichowsky
众所周知，均值-方差投资组合选择是一个时间不一致的最优控制问题，因为它不满足Bellman的最优性原理，因此通常的动态规划方法失败。我们开发了一个时间一致的制定这个问题，这是基于当地的最优性的概念，称为当地的均值方差效率，在一般的半鞅设置。我们从离散时间开始，那里的公式很简单，然后找到连续时间的自然延伸。这补充和推广了Basak和Chabakauri（2010）的公式以及Björk和Murgoci（2010）的相应示例，其中处理和最优性的概念依赖于潜在的马尔可夫框架。我们证明了连续时间的制定表明，它符合连续时间的限制的离散时间的制定。这种收敛性的证明是基于局部最优策略的结构条件和Föllmer-Schweizer分解的均值-方差权衡的全局描述。作为副产品，这也给出了Föllmer-Schweizer分解的新收敛结果，即，局部风险最小化策略。
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 Björk 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 Föllmer–Schweizer decomposition of the mean-variance trade-off. As a by-product, this also gives new convergence results for the Föllmer–Schweizer decomposition, i.e., for locally risk-minimising strategies.