%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