%0 Journal Article
%T Multivariate transient price impact and matrix-valued positive definite functions
%+ Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS)
%+ Mathematical Risk Handling (MATHRISK)
%+ Department of Mathematics and Computer Science [Mannheim]
%A Alfonsi, Aurélien
%A Schied, Alexander
%A Klöck, Florian
%< avec comité de lecture
%@ 0364-765X
%J Mathematics of Operations Research
%I INFORMS
%8 2016-03-01
%D 2016
%Z 1310.4471
%R 10.1287/moor.2015.0761
%K Multivariate price impact
%K matrix-valued positive de nite function
%K optimal trade execution
%K optimal portfolio liquidation
%K matrix function
%Z Mathematics [math]/Optimization and Control [math.OC]
%Z Quantitative Finance [q-fin]/Trading and Market Microstructure [q-fin.TR]Journal articles
%X 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.
%G English
%L hal-00919895
%U https://enpc.hal.science/hal-00919895
%~ ENPC
%~ INRIA
%~ INRIA-ROCQ
%~ CERMICS
%~ AO-ECONOMIE
%~ INSMI
%~ PARISTECH
%~ TESTALAIN1
%~ INRIA2
%~ TDS-MACS
%~ UNIV-EIFFEL
%~ UPEM-UNIVEIFFEL
%~ INRIAARTDOI
%~ INRIA-ALLEMAGNE