多元瞬时价格冲击与矩阵值正定函数
Multivariate transient price impact and matrix-valued positive definite functions
摘要：
我们考虑了一个模型的线性瞬态价格影响的多个资产，考虑跨资产的影响。我们的主要目标是挑选出需要施加在衰变核上的属性，以便模型允许行为良好的最优交易执行策略。我们首先表明，这种策略的存在性是保证假设衰变核对应于一个矩阵值正定函数。然而，一个例子表明，正定性本身并不能保证最优策略是良好的。基于一维情形的已有结果，我们研究了一类非增、非负、凸的衰减核，其值在对称$K\times K$矩阵中.我们发现，这些衰变核总是正定的，当它们甚至是严格正定的，一个结果，可能是独立的利益。当要求衰变核也是可交换的时，这类核的最优策略表现良好。我们展示了如何通过矩阵函数构建这种衰变核，并提供了一些例子。特别是我们完全解决了矩阵指数衰减的情况。
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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