英文标题：
《Time consistency of the mean-risk problem》
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作者：
Gabriela Kov\\\'a\\v{c}ov\\\'a, Birgit Rudloff
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最新提交年份：
2020
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英文摘要：
  Choosing a portfolio of risky assets over time that maximizes the expected return at the same time as it minimizes portfolio risk is a classical problem in Mathematical Finance and is referred to as the dynamic Markowitz problem (when the risk is measured by variance) or more generally, the dynamic mean-risk problem. In most of the literature, the mean-risk problem is scalarized and it is well known that this scalarized problem does not satisfy the (scalar) Bellman\'s principle. Thus, the classical dynamic programming methods are not applicable. For the purpose of this paper we focus on the discrete time setup, and we will use a time consistent dynamic convex risk measure to evaluate the risk of a portfolio. We will show that when we do not scalarize the problem, but leave it in its original form as a vector optimization problem, the upper images, whose boundary contains the efficient frontier, recurse backwards in time under very mild assumptions. Thus, the dynamic mean-risk problem does satisfy a Bellman\'s principle, but a more general one, that seems more appropriate for a vector optimization problem: a set-valued Bellman\'s principle. We will present conditions under which this recursion can be exploited directly to compute a solution in the spirit of dynamic programming. Numerical examples illustrate the proposed method. The obtained results open the door for a new branch in mathematics: dynamic multivariate programming.
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中文摘要：
随着时间的推移，选择一个在最大化预期收益的同时又将投资组合风险降至最低的风险资产组合是数学金融学中的一个经典问题，被称为动态马科维茨问题（当风险用方差衡量时）或更一般的动态平均风险问题。在大多数文献中，平均风险问题是标量化的，众所周知，这个标量化问题不满足（标量）Bellman原理。因此，经典的动态规划方法不适用。为了本文的目的，我们关注离散时间设置，并将使用时间一致的动态凸风险度量来评估投资组合的风险。我们将表明，当我们不将问题规模化，而是将其保留为向量优化问题的原始形式时，其边界包含有效边界的上部图像在非常温和的假设下会在时间上向后递归。因此，动态平均风险问题确实满足Bellman原理，但更一般的是，它似乎更适合向量优化问题：集值Bellman原理。我们将提出一些条件，在这些条件下，可以直接利用这种递归来计算符合动态规划精神的解。数值算例说明了该方法的有效性。所得结果为数学中的一个新分支打开了大门：动态多元规划。
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分类信息：

一级分类：Quantitative Finance        数量金融学
二级分类：Mathematical Finance        数学金融学
分类描述：Mathematical and analytical methods of finance, including stochastic, probabilistic and functional analysis, algebraic, geometric and other methods
金融的数学和分析方法，包括随机、概率和泛函分析、代数、几何和其他方法
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