https://enpc.hal.science/hal-00919895Alfonsi, AurélienAurélienAlfonsiCERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique - ENPC - École des Ponts ParisTechMATHRISK - Mathematical Risk Handling - UPEM - Université Paris-Est Marne-la-Vallée - ENPC - École des Ponts ParisTech - Inria de Paris - Inria - Institut National de Recherche en Informatique et en AutomatiqueSchied, AlexanderAlexanderSchiedDepartment of Mathematics and Computer Science [Mannheim] - Universität MannheimKlöck, FlorianFlorianKlöckDepartment of Mathematics and Computer Science [Mannheim] - Universität MannheimMultivariate transient price impact and matrix-valued positive definite functionsHAL CCSD2016Multivariate price impactmatrix-valued positive de nite functionoptimal trade executionoptimal portfolio liquidationmatrix function[MATH.MATH-OC] Mathematics [math]/Optimization and Control [math.OC][QFIN.TR] Quantitative Finance [q-fin]/Trading and Market Microstructure [q-fin.TR]Alfonsi, Aurélien2013-12-17 14:49:582023-06-26 04:18:292013-12-17 14:49:58enJournal articles10.1287/moor.2015.07611We 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.