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Article Dans Une Revue Mathematics of Operations Research Année : 2016

## Multivariate transient price impact and matrix-valued positive definite functions

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Aurélien Alfonsi

#### Résumé

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.

### Dates et versions

hal-00919895 , version 1 (17-12-2013)

### Identifiants

• HAL Id : hal-00919895 , version 1
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• DOI :

### Citer

Aurélien Alfonsi, Alexander Schied, Florian Klöck. Multivariate transient price impact and matrix-valued positive definite functions. Mathematics of Operations Research, 2016, ⟨10.1287/moor.2015.0761⟩. ⟨hal-00919895⟩

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