英文标题：
《Multivariate transient price impact and matrix-valued positive definite
functions》
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作者：
Aur\\\'elien Alfonsi, Alexander Schied, Florian Kl\\\"ock
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最新提交年份：
2015
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英文摘要：
We consider a model for linear transient price impact for multiple assets that takes cross-asset impact into account. Our main goal is to single out properties that need to be imposed on the decay kernel so that the model admits well-behaved optimal trade execution strategies. We first show that the existence of such strategies is guaranteed by assuming that the decay kernel corresponds to a matrix-valued positive definite function. An example illustrates, however, that positive definiteness alone does not guarantee that optimal strategies are well-behaved. Building on previous results from the one-dimensional case, we investigate a class of nonincreasing, nonnegative and convex decay kernels with values in the symmetric $K\\times K$ matrices. We show that these decay kernels are always positive definite and characterize when they are even strictly positive definite, a result that may be of independent interest. Optimal strategies for kernels from this class are well-behaved when one requires that the decay kernel is also commuting. We show how such decay kernels can be constructed by means of matrix functions and provide a number of examples. In particular we completely solve the case of matrix exponential decay.
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中文摘要：
我们考虑多个资产的线性瞬时价格影响模型，该模型考虑了交叉资产影响。我们的主要目标是挑出需要施加在衰变核上的属性，以便该模型允许行为良好的最优交易执行策略。我们首先通过假设衰变核对应于一个矩阵值正定函数，证明了这种策略的存在性。然而，一个例子说明，仅仅是正定性并不能保证最优策略的良好表现。基于一维情形的先前结果，我们研究了一类非递增、非负和凸的衰变核，其值在对称的$K \\乘以K$矩阵中。我们证明了这些衰变核总是正定的，并且当它们甚至是严格正定的时候，它们都具有特征，这一结果可能是独立的。当需要衰变核也是交换核时，此类核的最优策略表现良好。我们展示了如何利用矩阵函数构造这样的衰变核，并给出了一些例子。特别地，我们完全解决了矩阵指数衰减的情况。
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分类信息：

一级分类：Quantitative Finance 数量金融学
二级分类：Trading and Market Microstructure 交易与市场微观结构
分类描述：Market microstructure, liquidity, exchange and auction design, automated trading, agent-based modeling and market-making
市场微观结构，流动性，交易和拍卖设计，自动化交易，基于代理的建模和做市
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一级分类：Mathematics 数学
二级分类：Optimization and Control 优化与控制
分类描述：Operations research, linear programming, control theory, systems theory, optimal control, game theory
运筹学，线性规划，控制论，系统论，最优控制，博弈论
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